$(\Upsilon_{2}^{\mu\nu},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}x)\cong(\Upsilon_{2}^{\nu\mu},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2}{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}),\quad(\Upsilon_{2}^{\mu\nu},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4})\cong(\Upsilon_{2}^{\nu\mu},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2}{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4}),\quad(\Upsilon_{2}^{\mu\nu},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v})\cong(\Upsilon_{2}^{\nu\mu},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2}{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v})\qquad(\textrm{via
}\varrho)$

(see Figure 4.19), so that only the cases
$e={\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$
and
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}\in
e$ remain. The edge
$e={\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2b}$
can be taken care of by $\tau_{\nu}$-symmetry, because

$(\Upsilon_{2}^{\mu
0},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2}{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b})\cong(\Upsilon_{2}^{\mu
0},{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}1a})\quad\text{
and }\quad(\Upsilon_{2}^{\mu
1},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2}{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b})\cong(\Upsilon_{2}^{\mu
1},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4c})\,.$

Next, we have
$(\Upsilon_{2}^{\mu\nu},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u})\cong(\Upsilon_{2}^{\nu\mu},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2}{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b})$
(via $\varrho$) and, finally, the other edges containing
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$
are $\tau_{\nu}$-equivalent to certain edges containing
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$
or
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$.
∎

###  4.6. The attachment lemma

In this subsection we establish the attachment lemma for graphs in
$\mathfrak{D}_{4}(\Upsilon_{i}^{\mu\nu})$, which allows us to complete the
proof of Theorem 1.3( ) ‣ 1.3.

######  Lemma 4.29 (Attachment Lemma).

If $\Upsilon_{i}^{\mu\nu}\subseteq
G\in\mathfrak{D}_{4}(\Upsilon_{i}^{\mu\nu})$, then every vertex $q\in V(G)$ is
an $\Upsilon_{i}^{\mu\nu}$-twin of some vertex belonging to
$\Upsilon_{i}^{\mu\nu}$.

###### Proof.

Instead of “$\Upsilon_{i}^{\mu\nu}$-twin” we will just write “twin” throughout
the argument. We proceed with fourteen claims of the form that if a given
vertex $q\in V(G)$ is adjacent to certain members of
$V(\Upsilon_{i}^{\mu\nu})$, then $q$ is indeed a twin of some vertex of
$\Upsilon_{i}^{\mu\nu}$.

######  Claim 4.30.

If $q$ is adjacent to at least two vertices among
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$,
then either it is a twin of one of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$
or $\mu=0$ and $q$ is a twin of $y$.

###### Proof.

Suppose first that
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}\in\mathrm{N}(q)$.
With an arbitrary red auxiliary path $\pi$ we form the graph $\Upsilon(\pi)$
(see Figure 4.20(a)). The beautiful lemma yields $x\in\mathrm{N}(q)$; so if
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}\in\mathrm{N}(q)$
holds as well, then $q$ is a twin of $y$ (and $\mu=0$). Otherwise Lemma 4.10
shows that $q$ is a $\Upsilon(\pi)$-twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
which means that $q$ is adjacent to the end vertices of $\pi$. As for every
red vertex
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}\in{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\Gamma_{\mathrm{red}}}$
there is an auxiliary red path starting in
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}$,
we conclude that if
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}q\not\in
E(G)$, then
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\Gamma_{\mathrm{red}}}\subseteq\mathrm{N}(q)$,
whence $q$ is a twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$.
By $\tau_{\nu}$-symmetry the only other case we need to consider is that
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}\in\mathrm{N}(q)$
but
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}\not\in\mathrm{N}(q)$.
If $i\geq 3$ we can simply repeat the above argument with blue auxiliary
paths, thereby learning that $q$ is a twin of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$.

${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$$x$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}1}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}$
(a)
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}})$

${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$$x$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j+i-1}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}j+2i-1}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j-1}$
(b)
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j-1}})$

${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$$x$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}j+i-1}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j-i}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j-1}$
(c)
$\Upsilon(\pi_{{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j-1}})$

Figure 4.23. The proof of Claim 4.30 and Claim 4.33.

If $i=2$, however, only the red auxiliary path
$\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}}={\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}-{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2}-{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4}-{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}1}$
might be available. Corollary 4.24 guarantees that
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$
is reliable with respect to
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}})$
(see Figure 4.23(a)). Thus Lemma 4.11 applied to this graph yields
$x\in\mathrm{N}(q)$ (see Figure 4.12(c)). Next Lemma 4.12 and
$q\not\in\mathrm{Ext}(\Upsilon(\pi_{0}),x)$ imply that some
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}})$-twin
$4^{\prime}$ of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4}$
is in $\mathrm{N}(q)$ (see Figure 4.12(d)). Due to $i=2$ this vertex is
actually a real twin of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4}$.
Now $q$ is a $\Upsilon_{2}^{\mu\nu}(4^{\prime})$-twin of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$.
By Lemma 2.3 it follows that $q$ is a twin of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$.
∎

In view of $\sigma$-symmetry, the previous claim implies the following.

######  Claim 4.31.

If $\mu=0$ and $q$ is adjacent to at least two among
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$,
then it is a twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$,
or $x$. ∎

Without the assumption on $\mu$ we still have the following weaker assertion.

######  Claim 4.32.

If $q$ is adjacent to two vertices among
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$
and to some vertex from $\Gamma_{i}$, then $q$ is a twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$,
or
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$.

###### Proof.

Arguing indirectly we assume that $q$ is a counterexample. Let
$\varphi\in\\{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\mathrm{red}},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}\mathrm{green}},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}\mathrm{blue}}\\}$
be a colour satisfying $\mathrm{N}(q)\cap\Gamma_{\varphi}\neq\varnothing$.
Claim 4.31 tells us that $\mu=1$, whence
$\Gamma_{\varphi}\not\subseteq\mathrm{N}(q)$. Consequently, there is a pair of
consecutive vertices $\\{j,j+1\\}\subseteq\Gamma_{\varphi}$ such that exactly
one of $j$, $j+1$ is in $\mathrm{N}(q)$.

Recall that

$\pi_{j}=j-(j+i)-(j+2i)-(j+1)$

is an auxiliary path of colour $\varphi$. By Lemma 4.11 applied to
$\Upsilon(\pi_{j})$ there is a neighbour of $q$ in
$\mathrm{Ext}(\Upsilon(\pi_{j}),s)$ for some
$s\in\\{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}\\}$
whose colour is not $\varphi$ (see Figure 4.20). But this contradicts the fact
that $s$ is, by Corollary 4.24, reliable in $\Upsilon(\pi_{j})$. ∎

Here is a statement that can be viewed as an adaptation of Claim 3.5 to Vega
graphs.

######  Claim 4.33.

If
$j\neq{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}$
and $j,j+i-1\in\mathrm{N}(q)$, then $q$ is a twin of $j-i$.

###### Proof.

The condition
$j\neq{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}$
just means that $j$, $j+i-1$ have distinct colours. So by
$\tau_{\nu}$-symmetry we may assume that $j$ is red or green. If
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}$
is red, we consider
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j-1}})$,
where

$\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j-1}}={\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}(j-1)}-{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}(j+i-1)}-{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}(j+2i-1)}-{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}$

is a red auxiliary path (see Figure 4.23(b)). Assume for the sake of
contradiction that
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}\not\in\mathrm{N}(q)$.
In view of Lemma 4.11 there exists some
$y^{\prime}\in\mathrm{N}(q)\cap\mathrm{Ext}(\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j-1}}),x)$.
As $y^{\prime}$ is adjacent to
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$,
and $x$, we have $\mu=0$ and $y^{\prime}$ is a twin of $y$. But now $q$ has a
small neighbourhood in $\Upsilon_{i}^{\mu\nu}(y^{\prime})$, which is absurd.
This proves
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}\in\mathrm{N}(q)$.

Using the impossibility of small neighbourhoods again we obtain
$x\not\in\mathrm{N}(q)$. Therefore Lemma 4.12 shows that some
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j-1}})$-twin
$c^{\prime}$ of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$
is in $\mathrm{N}(q)$. Claim 4.30 informs us that $c^{\prime}$ is, in fact, a
real twin of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$.
So Lemma 4.19 applied to $\Upsilon_{i}^{\mu\nu}(c^{\prime})$ yields
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}[j,i-1]}\subseteq\mathrm{N}(q)$
and
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}[i,j+i-1]}\subseteq\mathrm{N}(q)$.
Altogether $q$ is a $\Upsilon_{i}^{\mu\nu}(c^{\prime})$-twin of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}j-i}$
and by Lemma 2.3 we are done. The case that
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j}$
is green is similar, the only difference being that now the path
$\pi_{{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j-1}}={\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}(j-1)}-{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}(j+i-1)}-{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}(j-i)}-{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j}$
is green (see Figure 4.23(c)). ∎

The next three claims analyse vertices adjacent to two opposite vertices of
the external hexagon.

######  Claim 4.34.

If
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a},{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}\in\mathrm{N}(q)$,
then $q$ is a twin of some vertex belonging to
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\Gamma_{\mathrm{red}}}$.

###### Proof.

Due to Lemma 4.20 it suffices to show the following statements.

  1. (0 )

For every
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}\in{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}[0,i-2]}$,
either
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j+i}$
or
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}j+2i}$
is in $\mathrm{N}(q)$.

  2. (0 )

If $\nu=0$, then
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2i-1}\in\mathrm{N}(q)$.

For the proof of (0 ) we apply Corollary 4.9 to
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}})$,
where, as usual,

$\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}}={\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}-{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}(j+i)}-{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}(j+2i)}-{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}(j+1)}$

(see Figures 4.12(a) and 4.24(a)). This shows that if
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j+i},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}j+2i}\not\in\mathrm{N}(q)$,
then there are
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}})$-twins
of
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j+i}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}j+2i}$
adjacent to $q$. But by Claim 4.33 these twins had to be real twins and the
twin lemma would show that they are adjacent, thus creating a triangle with
$q$.

To verify (0 ) we work with the green auxiliary path
$\omega={\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}-{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i}-{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}(i-1)}-{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}(2i-1)}$.
In view of Lemma 4.25 no vertex is adjacent to
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2i-1}$
and thus
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}$
is reliable in $\Upsilon(\omega)$ (see Figure 4.24(b)). So Lemma 4.11
discloses that
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}(2i-1)}\not\in\mathrm{N}(q)$
is impossible. ∎

${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$$x$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j+i}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}j+2i}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j+1}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}$
(a)
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}})$

${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$$x$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}i-1}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2i-1}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}$
(b) $\Upsilon(\omega)$

${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$$x$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i+j}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i+j}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i+j-1}$
(c)
$\Upsilon(\pi_{{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i+j-1}})$

${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$$x$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}i-1}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}$
(d) $\Upsilon(\eta)$

Figure 4.24. The proofs of Claim 4.34 and Claim 4.35.

######  Claim 4.35.

If
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}\in\mathrm{N}(q)$,
then $q$ is a twin of some vertex in
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}\Gamma_{\mathrm{blue}}}$.

###### Proof.

Because of Lemma 4.19 it suffices to show that

  1. (0 )

for every
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}\in{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}[1,i-2]}$,
either
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}$
or
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}j+i}$
is in $\mathrm{N}(q)$;

  2. (0 )

${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}i-1}\in\mathrm{N}(q)$;

  3. (0 )

and
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}\in\mathrm{N}(q)$.

The argument establishing (0 ) is very similar to the previous proof but uses
the blue auxiliary path

$\pi_{{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i+j-1}}={\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}(2i+j-1)}-{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}-{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}(i+j)}-{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}(2i+j)}$

instead (see Figure 4.24(c)); we omit the details.

For the remaining two statements we work with the red auxiliary path

$\eta={\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}-{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}-{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i}-{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}(i-1)}$

(see Figure 4.24(d)). Assume first that contrary to (0 ) we have
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}i-1}\not\in\mathrm{N}(q)$.
By Lemma 4.11 there exists some
$z\in\mathrm{N}(q)\cap\mathrm{Ext}(\Upsilon(\omega),{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0})$.
Due to
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v},{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}i-1}\in\mathrm{N}(z)$
and Lemma 4.17( ) ‣ 4.17 we have
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\Gamma_{\mathrm{red}}}\subseteq\mathrm{N}(z)$,
wherefore $z$ is a twin of
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2i-1}$
(and $\nu=0$). But this means that $q$ violates Lemma 4.25 with respect to
$\Upsilon_{i}^{\mu\nu}(z)$. Thereby (0 ) is proved.

Suppose next that
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}\not\in\mathrm{N}(q)$
and observe that due to Lemma 4.25 the vertex
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}i-1}$
is reliable with respect to $\Upsilon(\eta)$. So Lemma 4.12 tells us that
$\mathrm{N}(q)$ contains some $\Upsilon(\eta)$-twins $0^{\prime}$,
$i^{\prime}$ of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}$,
respectively. By Lemma 4.20( ) ‣ 4.20 $0^{\prime}$ is actually a real twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}$
and by the case
$j={\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i}$
of Claim 4.33 $i^{\prime}$ is also real twin of
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}$.
Now the twin lemma discloses $0^{\prime}i^{\prime}\in E(G)$ and together with
$q$ this edge closes a triangle, which is absurd. This concludes the proof of
(0 ). ∎

######  Claim 4.36.

If
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}\in\mathrm{N}(q)$,
then $q$ is a twin of some vertex in
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}\Gamma_{\mathrm{green}}}$.

###### Proof.

By $\tau_{\nu}$-symmetry this follows from the two previous claims. ∎

So far all our claims assume that at least two neighbours of $q$ in
$V(\Upsilon^{\mu\nu}_{i})$ are given. This is not surprising, because all the
results in §4.3 are of this form, and up to this point no other arguments have
been utilised. Eventually we need to cover less restrictive cases of the
attachment lemma as well. Accordingly, we shall use a hexagon argument in the
claim after the next one, where we show that neighbours of $x$ are twins of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$,
or $y$. Preparing ourselves for this task we establish an important special
case first.

######  Claim 4.37.

If $q$ is adjacent to $x$ and a vertex belonging to $\Gamma_{i}$, then it is a
twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
or
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$.

###### Proof.

If, for instance,
$x,{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}\in\mathrm{N}(q)$,
where
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}\in{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\Gamma_{\mathrm{red}}}$,
we take a red auxiliary path $\pi$ one of whose end vertices is $j$ and apply
Lemma 4.10 to $\Upsilon(\pi)$ (see Figure 4.20(a)). As $q$ cannot have a small
neighbourhood in $\Upsilon_{i}^{\mu\nu}$, this yields
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}\in\mathrm{N}(q)$
and due to Claim 4.30 $q$ is a twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$.
By $\tau_{\nu}$-symmetry the only other possibility we need to consider is
that
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}j},x\in\mathrm{N}(q)$
holds for some
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}j}\in{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}\Gamma_{\mathrm{blue}}}}$.
The case $i\geq 3$ is similar, because then there is a blue auxiliary path
starting in
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}j}$.
Suppose, finally, that $i=2$ and
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4},x\in\mathrm{N}(q)$.
Since
$\varrho({\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4})={\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4}$
and
$\varrho(x)={\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}$,
the case
$j={\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}4}$
of Claim 4.33 now shows that $q$ is a twin of
$\varrho^{-1}({\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}2})$,
i.e., of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$.
∎

######  Claim 4.38.

If $x\in\mathrm{N}(q)$, then $q$ is a twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}b}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}c}$,
or $y$.

###### Proof.

We first show that if
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}\not\in\mathrm{N}(q)$,
then $q$ is a twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$.
To this end we take an arbitrary common neighbour $z$ of $q$,
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$
and form the hexagon shown in Figure 4.25(a). Since $G$ satisfies
$\mathcal{D}_{2}$, there is a common neighbour $t$ of
$\\{q,{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a},{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}\\}$
or
$\\{x,{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0},z\\}$.

If
$q,{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a},{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}\in\mathrm{N}(t)$,
then Claim 4.34 informs us that $t$ is a twin of some
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}\in{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\Gamma_{\mathrm{red}}}$.
Among the neighbours of $q$ in $\Upsilon_{i}^{\mu\nu}(t)$ there are $x$ and
the red vertex
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}t}\in{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\Gamma_{\mathrm{red}}}$.
Therefore, Claim 4.37 implies that $q$ is a $\Upsilon_{i}^{\mu\nu}(t)$-twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$
and Lemma 2.3 reveals that $q$ is a real twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$
as well.

$q$$x$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$$z$
(a)

$q$$x$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$$z$
(b)

$q$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$$z$
(c)

$q$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$$z$
(d)

$q$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$$z$
(e)

Figure 4.25. Hexagon arguments.

Suppose next that
$x,{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0},z\in\mathrm{N}(t)$.
By Claim 4.37, $t$ is a twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$.
For simplicity we may assume
$t={\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$.
Now Claim 4.34 shows that $z$ is a twin of some
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}j}\in{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\Gamma_{\mathrm{red}}}$
and a final application of Claim 4.37 reveals that $q$ is again a twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$.
Summarising the discussion so far, we have thereby completed the case
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}\not\in\mathrm{N}(q)$.

A similar argument based on the hexagon in Figure 4.25(b) shows that if
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}\not\in\mathrm{N}(q)$,
then $q$ is a twin of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$.
In the only remaining case,
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}\in\mathrm{N}(q)$,
Claim 4.30 implies that $q$ is a twin of
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$
or $y$. ∎

This has the following consequence.

######  Claim 4.39.

If $\mu=0$ and some twin $y^{\prime}$ of $y$ is adjacent to $q$, then $q$ is a
twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}v}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}w}$,
or $x$.

###### Proof.

For $y=y^{\prime}$ this can be seen by applying the automorphism $\sigma$ to
the previous claim. By Lemma 2.3 the general case follows. ∎

Our next major goal is an attachment lemma for neighbours of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$
(cf. Claim 4.41) and again we commence with a special case.

######  Claim 4.40.

If
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}\in\mathrm{N}(q)$
and, moreover,
$\\{{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b},{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}\\}\cap\mathrm{N}(q)\neq\varnothing$,
then $q$ is a twin of
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$,
or $x$.

###### Proof.

We will only display the argument for the case
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}\in\mathrm{N}(q)$,
the other case being similar. Using the red auxiliary path
$\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}}={\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}-{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}i}-{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i}-{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}1}$
we form the graph
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}})$.
By Corollary 4.9 there is a
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}})$-twin
of $c$ or
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}2i}$
adjacent to $q$.

Suppose first that $c^{\prime}\in\mathrm{N}(q)$ holds for some
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}})$-twin
$c^{\prime}$ of $c$. Claim 4.30 tells us that $c^{\prime}$ is a real twin of
$c$ and due to Lemma 4.16( ) ‣ 4.16 $q$ is a
$\Upsilon^{\mu\nu}_{i}(c^{\prime})$-twin of $x$. So by Lemma 2.3 $q$ is a twin
of $x$.

It remains to consider the case that some
$\Upsilon(\pi_{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}0}})$-twin
$(2i)^{\prime}$ of $2i$ is adjacent to $q$. For
${j={\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}1}}$
Claim 4.33 shows that $(2i)^{\prime}$ is actually a real twin of $2i$ and,
therefore, Claim 4.32 is applicable to $\Upsilon^{\mu\nu}_{i}((2i)^{\prime})$.
Together with Lemma 2.3 we conclude that $q$ is a twin of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$.
∎

######  Claim 4.41.

If $q$ is adjacent to
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
then it is a twin of some vertex of $\Upsilon_{i}^{\mu\nu}$.

###### Proof.

If
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}\in\mathrm{N}(q)$
the desired conclusion can be drawn from Claim 4.34. Assuming
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}\not\in\mathrm{N}(q)$
from now on we take a common neighbour $z$ of $q$,
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$
and form the hexagon shown in Figure 4.25(c).

If there is a common neighbour $t$ of $q$,
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$,
then Claim 4.30 shows that $t$ is a twin of
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
or $\mu=0$ and $t$ is a twin of $y$. In the latter case we use Claim 4.39 and
in the former case we appeal to Claim 4.40 and Lemma 2.3.

Since $G$ satisfies $\mathcal{D}_{2}$, it only remains to consider the case
that some common neighbour $t$ of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
$z$ exists. By Claim 4.40, $t$ is a twin of
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$
or $x$. Due to Lemma 2.3 we can assume, for simplicity, that
$t\in\\{{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w},x\\}$.
Now the Claims 4.30 and 4.38 tell us that $z$ is a twin of
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$
or $y$. Finally, Claim 4.39 or Claim 4.40 and Lemma 2.3 complete the proof. ∎

We also need a version of this claim with
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$
or
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$
instead of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
which we prepare as follows.

######  Claim 4.42.

If $q$ is adjacent to
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$
and
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$,
then it is a twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$
or $x$.

###### Proof.

If
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}\in\mathrm{N}(q)$,
then Claim 4.41 shows that $q$ is a twin of $x$. Otherwise, we take a common
neighbour $z$ of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
$q$ and deduce from Claim 4.41 that $z$ is a twin of some vertex $t\in
V(\Upsilon^{\mu\nu}_{i})$ adjacent to
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$.
Since neither
$q{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}z$
nor
$q{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}z$
is a triangle in $G$, we have $z\not\in(\mathrm{N}(b)\cup\mathrm{N}(c))$.
Consequently, $t$ is non-adjacent to
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$
and
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$
and, altogether, only the possibility
$t\in{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}\Gamma_{\mathrm{red}}}$
remains. Now Claim 4.32 shows that $q$ is a $\Upsilon^{\mu\nu}_{i}(t)$-twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$
and another application of Lemma 2.3 concludes the argument. ∎

Next we can repeat the proof of Claim 4.41 with the hexagon in Figure 4.25(d)
or 4.25(e), thus obtaining the following statement.

######  Claim 4.43.

If
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$
or
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$
is in $\mathrm{N}(q)$, then $q$ is a twin of some vertex of
$\Upsilon_{i}^{\mu\nu}$. ∎

Now the attachment lemma is clear. Given any $q\in V(G)$ we can either apply
Claim 4.38 directly (if $qx\in E(G)$), or there is a common neighbour $z$ of
$q$ and $x$, which then has to be a twin of
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$,
or $y$. As usual, Lemma 2.3 allows us to assume that, actually, $z$ is one of
those four vertices. Depending on $z$ we now use Claim 4.41, 4.43, or 4.39. ∎

Let us end this section by pointing out that owing to the twin lemma (cf.
Lemma 4.28) and the attachment lemma (cf. Lemma 4.29) Lemma 2.4 implies
Theorem 1.3( ) ‣ 1.3.

##  5\. Concluding remarks

###  5.1. Finite graphs

Whenever we appealed to the property $\mathcal{D}_{4}$ in the proof of Theorem
1.2, the list of $3m$ vertices $x_{1},\dots,x_{3m}$ we specified had no
independent subset $\\{x_{i}\colon i\in I\\}$ such that $|I|\geq m+2$.
Originally we thought that our intended application to Ramsey-Turán theory,
i.e., the proof of (1.1), required that we establish Theorem 1.2 with such
‘restricted applications’ of $\mathcal{D}_{4}$ only. While it turned out later
that this extra caution could be avoided, we would still like to record the
stronger statement for potential future references.

Let us say for a positive integer $k$ that a graph $G$ has the property
$\mathcal{Q}_{k}$ if for every $m\in[k]$ and every sequence
$x_{1},\dots,x_{3m}$ of vertices of $G$ there is an index set $I\subseteq[3m]$
such that $U=\\{x_{i}\colon i\in I\\}$ is independent and either $|I|\geq
m+2$, or $|I|=m+1$ and $U$ has a common neighbour.

######  Theorem 5.1.

A maximal triangle-free graph satisfies $\mathcal{Q}_{4}$ if and only if it is
a blow-up of either an Andrásfai or a Vega graph. ∎

Recall that by Lemma 4.2 the members of $\mathfrak{D}_{4}$ contain no induced
cubes. We also found several other forbidden induced subgraphs for the class
$\mathfrak{D}_{4}$, such as the graph $N$ depicted in Figure 4.5, but so far
we did not complete our analysis of the situation.

######  Conjecture 5.2.

There exists a finite family $\mathscr{F}$ of graphs such that
$\mathfrak{D}_{4}$ is the class of maximal triangle-free graphs with at least
two vertices not possessing induced subgraphs in $\mathscr{F}$.

Let us point out that a somewhat similar result for the class $\mathfrak{A}$
of maximal triangle-free, $\Upsilon$-free graphs and the forbidden family
$\\{C_{6}\\}$ is established in Section 3. Thus a solution of the above
problem might lead to a different (albeit longer) proof of Theorem 1.2.

Another problem suggested by Theorem 1.2 is whether the assumption
$\mathcal{D}_{4}$ is really necessary or whether a more elaborate argument
would show that $\mathcal{D}_{3}$ suffices. It is probably not very difficult
to rule this out, but we lacked the energy for doing so.

######  Conjecture 5.3.

There is a finite maximal triangle-free graph $G$ satisfying $\mathcal{D}_{3}$
but not $\mathcal{D}_{4}$.

The last finitary problem we would like to mention is the well-known question
to characterise the maximal triangle-free graphs $G$ on $n$ vertices
satisfying $\delta(G)\geq n/3$. This family does not consist exclusively of
blow-ups of Andrásfai and Vega graphs, since it contains, for instance, the
Cayley graphs associated to

${\mathds{Z}}/6k{\mathds{Z}}\quad\text{and}\quad\\{\pm
k,\pm(k+1),\dots,\pm(2k-1)\\}.$

This graph contains the induced hexagon $0-2k-3k-4k-5k-0$ but no vertex with
three neighbours on this hexagon and, therefore, it violates
$\mathcal{D}_{2}$. Another graph interesting in this context is shown in
Figure 5.1 (see also Figure 4.16(c) for a less symmetric drawing of the same
graph). As all these additional examples are $(n/3)$-regular, one may wonder
whether the following is true.

######  Question 5.4.

Let $G$ be a maximal triangle-free graph on $n$ vertices such that
$\delta(G)\geq n/3$, but at least one vertex of $G$ has degree larger than
$n/3$. Does it follow that $G$ is a blow-up of either an Andrásfai or a Vega
graph?

Figure 5.1. A triangle-free graph $G$ with $v(G)=12$ and $\alpha(G)=4$.

###  5.2. Infinite graphs

One may wonder whether results analogous to Theorem 1.3 hold for infinite
graphs as well. Before we discuss this matter we present three examples of
infinite maximal triangle-free graphs, which have the property
$\mathcal{D}_{k}$ for every $k\geq 1$.

I. The circular Andrásfai graph $G_{\omega}$.

Let $\xi\in S^{1}\subseteq{\mathds{C}}$ be a complex number of modulus $1$
which is not a root of unity. Consider the graph $G_{\omega}$ with vertex set
$V=\\{\xi^{n}\colon n\in{\mathds{N}}\\}$ in which two vertices are adjacent if
their distance exceeds $\sqrt{3}$. Since $V$ is dense in $S^{1}$, the graph
$G_{\omega}$ is maximal triangle-free. Moreover, for every sequence of $3m$
vertices from $V$ there is an open arc $I\subseteq S^{1}$ of length $2\pi/3$
containing $m+1$ of them. Consequently, $G_{\omega}$ satisfies
$\mathcal{D}_{k}$ for every $k\geq 1$.

II. Generalised Andrásfai graphs $\Gamma_{X}$.

Given a dense linear order $(X,\leq)$ without minimal or maximal elements we
define the generalised Andrásfai graph $\Gamma_{X}$ to be the graph with
vertex set

$V(\Gamma_{X})=\\{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}r_{x}}\colon
x\in
X\\}\cup\\{{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}g_{x}}\colon
x\in
X\\}\cup\\{{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}b_{x}}\colon
x\in X\\}$

and all edges

  1. $\bullet$

${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}r_{x}}{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}g_{x}}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}g_{x}}{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}b_{x}}$,
where $x\in X$,

  2. $\bullet$

as well as
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}r_{x}}{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}g_{y}}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}g_{x}}{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}b_{y}}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}b_{x}}{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}r_{y}}$,
where $x<y$.

It can easily be checked that $\Gamma_{X}$ is maximal triangle-free and that
if $x_{1}<x_{2}<\dots<x_{k}$ are in $X$, then the vertices

$\\{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}r_{x_{1}}},{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}r_{x_{2}}},\dots,{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}r_{x_{k}}}\\}\cup\\{{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}g_{x_{1}}},{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}g_{x_{2}}},\dots,{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}g_{x_{k}}}\\}\cup\\{{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}b_{x_{1}}},\dots,{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}b_{x_{k-1}}}\\}$

span a copy of $\Gamma_{k}$ in $\Gamma_{X}$. This fact immediately implies
that $G_{X}$ has the property $\mathcal{D}_{k}$ for every $k\geq 1$.

III. Generalised Vega graphs $\Upsilon^{\mu 0}_{X}$.

Starting from the graph $\Gamma_{X}$ defined in the previous example we can
construct two generalised Vega graphs $\Upsilon^{00}_{X}$ and
$\Upsilon^{10}_{X}$. To this end we take an external hexagon
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$
and, as usual, we connect

  1. $\bullet$

${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$
to
$\\{{\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}r_{x}}\colon
x\in X\\}$;

  2. $\bullet$

${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$
to
$\\{{\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}g_{x}}\colon
x\in X\\}$;

  3. $\bullet$

and
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$
to
$\\{{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}b_{x}}\colon
x\in X\\}$.

Next, we join another vertex $x$ to
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}a}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}b}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}c}$,
thereby obtaining $\Upsilon^{10}_{X}$. Finally $\Upsilon^{00}_{X}$ has a
further vertex $y$ adjacent to
${\color[rgb]{0.75,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0,0}u}$,
${\color[rgb]{0,0.65,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.65,0}v}$,
${\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}w}$,
and $x$.

As none of these graphs is a blow-up of a finite Andrásfai of Vega graph, the
most naïve extension of Theorem 1.3 to infinite graphs is false.

######  Theorem 5.5.

There exists an infinite maximal triangle-free graph satisfying
$\mathcal{D}_{k}$ for every $k\geq 1$ that fails to be a blow-up of any finite
graph. ∎

We are optimistic, however, that the following ‘local version’ of Theorem 1.3
holds for infinite graphs.

######  Conjecture 5.6.

For every maximal triangle-free graph $G$ with property $\mathcal{D}_{4}$ and
every finite set of vertices $W\subseteq V(G)$ there exists a finite set $U$
with $W\subseteq U\subseteq V(G)$ which spans a blow-up of either an Andrásfai
or a Vega graph in $G$.

Notice that this is true for finite graphs, where one just needs to take
$U=V(G)$, and for the infinite graphs $G_{\omega}$, $\Gamma_{X}$,
$\Upsilon_{X}^{\mu 0}$ constructed above. Moreover, Conjecture 5.6 implies
that the following statement holds for $\ell=4$.

######  Conjecture 5.7.

There exists a natural number $\ell$ such that each maximal triangle-free
graph satisfying $\mathcal{D}_{\ell}$ has the property $\mathcal{D}_{k}$ for
every $k\geq 1$.

Next we would like to address the infinitary version of Conjecture 5.3. Let us
recall that Henson [He] constructed a countable homogeneous triangle-free
graph, which he denoted by $U_{3}$. Its main property is that for all disjoint
finite sets $A,B\subseteq V(U_{3})$ such that $A$ is independent in $U_{3}$
there is a vertex $v$ adjacent to the vertices in $A$ and non-adjacent to the
vertices in $B$. In particular, $U_{3}$ is a maximal triangle-free graph
containing all finite or countably infinite triangle-free graphs as induced
subgraphs.

######  Theorem 5.8.

Henson’s graph $U_{3}$ satisfies $\mathcal{D}_{3}$ but not $\mathcal{D}_{4}$.

###### Proof.

It is a well known elementary Ramsey theoretic fact that for all
$m\in\\{1,2,3\\}$ the partition relation $3m\longrightarrow(3,m+1)$ asserting
that triangle-free graphs on $3m$ vertices contain independent sets of size
$m+1$ holds. Consequently, for every sequence $x_{1},\dots,x_{3m}$ of vertices
of $U_{3}$ there is a set $I\subseteq[3m]$ of size $m+1$ such that
$A=\\{x_{i}\colon i\in I\\}$ is independent. As all finite independent subsets
of $V(U_{3})$ have common neighbours, this proves that $U_{3}$ satisfies
$\mathcal{D}_{3}$.

On the other hand, as $U_{3}$ contains the graph depicted in Figure 5.1, it
has twelve vertices no five of which possess a common neighbour. Thus $U_{3}$
does not have the property $\mathcal{D}_{4}$. ∎

Finally, we would like to point out that there is an infinite minimum-degree
version of the Brandt-Thomassé theorem. The result that follows concerns
graphs on ${\mathds{N}}$. Given such a graph $G$ and a vertex $v\in V(G)$ we
write

$\underline{\deg}(v)=\liminf_{n\to\infty}\frac{|\\{1,2,\dots,n\\}\cap\mathrm{N}(v)|}{n}\,.$

######  Theorem 5.9.

Let $G$ be a maximal triangle-free graph on ${\mathds{N}}$. If
$\inf\\{\underline{\deg}(v)\colon v\in{\mathds{N}}\\}>1/3$, then $G$ is a
blow-up of either an Andrásfai or a Vega graph.

###### Proof.

Choose $\varepsilon>0$ such that $\underline{\deg}(v)>1/3+\varepsilon$ holds
for every $v\in{\mathds{N}}$. Given any sequence $x_{1},\dots,x_{3m}$ of
vertices of $G$, there is some positive integer $n$ such that for every
$i\in[3m]$ we have $|\mathrm{N}(x_{i})\cap[n]|\geq(1/3+\varepsilon)n$. Now a
counting argument leads to some $y\in[n]$ for which the set
$I=\\{i\in[3m]\colon x_{i}y\in E(G)\\}$ has at least the size
$m+3m\varepsilon$. Consequently, $G$ satisfies $\mathcal{D}_{k}$ for every
$k\geq 1$ and, in particular, $G$ satisfies $\mathcal{D}_{4}$.

The only moment in the proof of Theorem 1.3 employing the finiteness of $G$ is
that at the very end we choose a maximal Andrásfai or Vega graph contained in
it. Thus it remains to show that there exists some constant $C$ such that
every Andrásfai or Vega subgraph of $G$ has at most $C$ vertices.

If for some $m\in{\mathds{N}}$ there is a copy of $\Gamma_{m+1}$ in $G$ and
$U=\\{x_{1},\dots,x_{3m}\\}$ contains $3m$ distinct vertices of this copy,
then every independent subset of $I\subseteq U$ satisfies
$|I|\leq\alpha(\Gamma_{m+1})=m+1$ and, therefore, the conclusion of our first
paragraph yields $m\leq(3\varepsilon)^{-1}$. This shows that the Andrásfai
subgraphs of $G$ are indeed bounded. For Vega graphs the argument is similar,
using in addition that each $\Upsilon^{\mu\nu}_{i}$ has for
$\kappa=9i-(6+\mu+\nu)$ a blow-up on $3\kappa-1$ vertices with independence
number $\kappa$ (see [BP] or [Vega]). ∎

## References


_actually_ produced the result. Similarly, as recently shown by Lundberg et
al. (2021), it is many times unclear what are authors _trying_ to estimate,
let alone under what assumptions they claim to do so. Providing a DAG in these
circumstances, rather than being “superfluous”, is critical to understand how
the authors see the mapping between their particular setting and the template
they are invoking, how identification works out in their particular
circumstances, and what changes can make it break down.

This bring us to the second point, that of _testability_. Graphical models
provide us with a method, based on $d-$separation, to _systematically_ derive
implications from a model, thus giving us the tools to test and falsify
scientific claims. It is a common joke to say that “robustness checks” are
useless since there is no published paper that has ever failed to pass one.
This is suggestive of the risks of cherry-picking _which_ implications are
being tested, and reinforces the need to focus instead on the implications _of
what_ are we testing: the underlying model, under which researchers claim to
identify a given causal effect414141One important take-away from this is the
need to treat quasi-experiments with the same level of scrutiny than
observational studies, especially with respect to covariate inclusion and
sensitivity to potential violations of identifying assumptions. This is
especially important in mixed or conditional strategies, where the target
template only works under some form of unconfoundedness. Many times, however,
that first step remains totally unattended, producing the paradoxical result
that the supposedly weakest link of a causal claim is subjected to a lesser
evidentiary standard than any analogous observational study would be.. The
same goes when assessing the credibility of empirical evidence; rather than
assuming a generally incredulous stance, critics should engage in producing
plausible alternative explanations that can be tested. As economist Claudia
Sahm has said, “be constructive, bring a model!”.

Finally, DAGs can be instrumental to improve the _generativity_ of our
scientific discussions, going beyond the vague skepticism that sometimes
characterizes our disagreements about empirical results. This is especially
important when, like it is the case in causal inference with observational
data, relying on untestable assumptions is unavoidable. Simply mentioning the
_possibility_ of unobserved confounding, for example, without providing a
plausible story on how it can be the case, does not help to move forward our
understanding. Using graphical models is just a formal and transparent way of
sharing our (contested) understanding about the world, which inevitably
informs our analyses. In this context, scientific criticism and theory
refinement is what occurs in the struggle to erase and redraw nodes and
arrows, to incorporate, omit, or break variables into more granular
mechanisms424242Even more, “complicating the picture”, in the sense of adding
new variables and arrows to a DAG, can be a fruitful exercise for theory
building, presenting new opportunities for identification, as highlighted by
Knight & Winship (2013). A similar argument can be found in Rohrer (2018),
where she proposes climate change as an example where the existence of a very
plausible mechanism is what increases our confidence in the inference being
made. Breaking down nodes and paths into more fine-grained processes helps to
clarify the role of hypothetical mechanisms, opening the (front-)door to new
identification opportunities and deriving additional testable implications
from the model.. Graphical models provide a principled tool for causal
reasoning that has ecumenical potential, across disciplines and ways of
theorizing, to guide these conversations.

## References

  * Angrist & Pischke (2009) Angrist, J. D., & Pischke, J.-S. (2009).  Mostly Harmless Econometrics: An Empiricist’s Companion.  Princeton University Press, 1st edition ed.
  * Angrist & Pischke (2010) Angrist, J. D., & Pischke, J.-S. (2010).  The Credibility Revolution in Empirical Economics: How Better Research Design is Taking the Con out of Econometrics.  Journal of Economic Perspectives, 24(2), 3–30.
  * Aronow & Sävje (2020) Aronow, P. M., & Sävje, F. (2020).  Review of The Book of Why: The New Science of Cause and Effect.  Journal of the American Statistical Association, 115(529), 482–485.
  * Athey & Imbens (2016) Athey, S., & Imbens, G. (2016).  Recursive partitioning for heterogeneous causal effects.  Proceedings of the National Academy of Sciences, 113(27), 7353–7360.
  * Bareinboim et al. (2020) Bareinboim, E., Correa, J. D., Ibeling, D., & Icard, T. (2020).  On Pearl’s Hierarchy and the Foundations of Causal inference.
  * Breen (2018) Breen, R. (2018).  Some Methodological Problems in the Study of Multigenerational Mobility.  European Sociological Review, 34(6), 603–611.
  * Brumback (2021) Brumback, B. A. (2021).  Fundamentals of Causal Inference: With R.  Boca Raton: Chapman and Hall/CRC, 1st edition ed.
  * Cartwright (1995) Cartwright, N. (1995).  Precis of Nature’s Capacities and Their Measurement.  Philosophy and Phenomenological Research, 55(1), 153.
  * Cartwright (2020) Cartwright, N. (2020).  No Causes in, No Causes out.  Los Angeles, (p. 43).
  * Cinelli et al. (2022) Cinelli, C., Forney, A., & Pearl, J. (2022).  A Crash Course in Good and Bad Controls.  Sociological Methods & Research, (p. 00491241221099552).
  * Cinelli & Hazlett (2020) Cinelli, C., & Hazlett, C. (2020).  Making sense of sensitivity: Extending omitted variable bias.  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 82(1), 39–67.
  * Cunningham (2021) Cunningham, S. (2021).  Causal Inference: The Mixtape.  New Haven ; London: Yale University Press.
  * Dawid (2000) Dawid, A. P. (2000).  Causal Inference without Counterfactuals.  Journal of the American Statistical Association, 95(450), 407–424.
  * Dawid (2020) Dawid, A. P. (2020).  Decision-theoretic foundations for statistical causality.  arXiv:2004.12493 [math, stat].
  * Doleac (2019) Doleac, J. L. (2019).  “Evidence-Based Policy” Should Reflect a Hierarchy of Evidence.  Journal of Policy Analysis and Management, 38(2), 517–519.
  * Eckles & Bakshy (2021) Eckles, D., & Bakshy, E. (2021).  Bias and High-Dimensional Adjustment in Observational Studies of Peer Effects.  Journal of the American Statistical Association, 116(534), 507–517.
  * Elwert (2013) Elwert, F. (2013).  Graphical Causal Models.  In S. L. Morgan (Ed.) Handbook of Causal Analysis for Social Research, Handbooks of Sociology and Social Research, (pp. 245–273). Dordrecht: Springer Netherlands.
  * Hausman & Rapson (2018) Hausman, C., & Rapson, D. S. (2018).  Regression Discontinuity in Time: Considerations for Empirical Applications.  Annual Review of Resource Economics, 10, 533–552.
  * Hernán & Robins (2023) Hernán, M. A., & Robins, J. M. (2023).  Causal Inference: What If.  Boca Raton: CRC Press.
  * Holland (1986) Holland, P. W. (1986).  Statistics and Causal Inference.  Journal of the American Statistical Association, 81(396), 945–960.
  * (21) Huntington-Klein, N. (????).  The Effect: An Introduction to Research Design and Causality | The Effect.
  * Imai et al. (2008) Imai, K., King, G., & Stuart, E. A. (2008).  Misunderstandings between experimentalists and observationalists about causal inference.  Journal of the Royal Statistical Society: Series A (Statistics in Society), 171(2), 481–502.
  * Imbens (2010) Imbens, G. W. (2010).  Better LATE Than Nothing: Some Comments on Deaton (2009) and Heckman and Urzua (2009).  Journal of Economic Literature, 48(2), 399–423.
  * Imbens (2020) Imbens, G. W. (2020).  Potential Outcome and Directed Acyclic Graph Approaches to Causality: Relevance for Empirical Practice in Economics.  Journal of Economic Literature, 58(4), 1129–1179.
  * Imbens & Rubin (1995) Imbens, G. W., & Rubin, D. B. (1995).  Discussion of Causal Diagrams for Empirical Research by J. Pearl.  Biometrika, 82(4), 694–695.
  * Imbens & Rubin (2015) Imbens, G. W., & Rubin, D. B. (2015).  Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction.  Cambridge University Press.
  * Knight & Winship (2013) Knight, C. R., & Winship, C. (2013).  The Causal Implications of Mechanistic Thinking: Identification Using Directed Acyclic Graphs (DAGs).  In S. L. Morgan (Ed.) Handbook of Causal Analysis for Social Research, Handbooks of Sociology and Social Research, (pp. 275–299). Dordrecht: Springer Netherlands.
  * Laubach et al. (2021) Laubach, Z. M., Murray, E. J., Hoke, K. L., Safran, R. J., & Perng, W. (2021).  A biologist’s guide to model selection and causal inference.  Proceedings of the Royal Society B: Biological Sciences, 288(1943), 20202815.
  * Levitt (2002) Levitt, S. D. (2002).  Using Electoral Cycles in Police Hiring to Estimate the Effects of Police on Crime: Reply.  American Economic Review, 92(4), 1244–1250.
  * Lewbel (2019) Lewbel, A. (2019).  The Identification Zoo: Meanings of Identification in Econometrics.  Journal of Economic Literature, 57(4), 835–903.
  * Liu et al. (2021) Liu, T., Ungar, L., & Kording, K. (2021).  Quantifying causality in data science with quasi-experiments.  Nature Computational Science, 1(1), 24–32.
  * Lundberg et al. (2021) Lundberg, I., Johnson, R., & Stewart, B. M. (2021).  What Is Your Estimand? Defining the Target Quantity Connects Statistical Evidence to Theory.  American Sociological Review, 86(3), 532–565.
  * Maclaren & Nicholson (2020) Maclaren, O. J., & Nicholson, R. (2020).  What can be estimated? Identifiability, estimability, causal inference and ill-posed inverse problems.  arXiv:1904.02826 [cs, math, stat].
  * Morgan & Winship (2014) Morgan, S. L., & Winship, C. (2014).  Counterfactuals and Causal Inference: Methods and Principles for Social Research.  Analytical Methods for Social Research. Cambridge University Press, 2nd edition ed.
  * Pearl (1995) Pearl, J. (1995).  Causal diagrams for empirical research.  Biometrika, 82(4), 669–688.
  * Pearl (2001) Pearl, J. (2001).  Direct and indirect effects.  In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, (pp. 411–420). San Francisco California USA: Morgan Kaufmann.
  * Pearl (2009) Pearl, J. (2009).  Causality.  Cambridge University Press.
  * Pearl (2010) Pearl, J. (2010).  The Foundations of Causal Inference.  Sociological Methodology, 40(1), 75–149.
  * Pearl et al. (2016) Pearl, J., Glymour, M., & Jewell, N. P. (2016).  Causal Inference in Statistics: A Primer.  Wiley.
  * Pearl & Mackenzie (2018) Pearl, J., & Mackenzie, D. (2018).  The Book of Why: The New Science of Cause and Effect.  Penguin Books Limited.
  * Pearl & Robins (1995) Pearl, J., & Robins, J. (1995).  Probabilistic evaluation of sequential plans from causal models with hidden variables.  UAI, 95, 444–453.
  * Peters et al. (2017) Peters, J., Janzing, D., & Schölkopf, B. (2017).  Elements of Causal Inference: Foundations and Learning Algorithms.  MIT Press.
  * Petersen & van der Laan (2014) Petersen, M. L., & van der Laan, M. J. (2014).  Causal Models and Learning from Data: Integrating Causal Modeling and Statistical Estimation.  Epidemiology, 25(3), 418–426.
  * Rauscher (2016) Rauscher, E. (2016).  Does Educational Equality Increase Mobility? Exploiting Nineteenth-Century U.S. Compulsory Schooling Laws.  American Journal of Sociology, 121(6), 1697–1761.
  * Robins et al. (2000) Robins, J. M., Hernán, M. Á., & Brumback, B. (2000).  Marginal Structural Models and Causal Inference in Epidemiology:.  Epidemiology, 11(5), 550–560.
  * Rohrer (2018) Rohrer, J. M. (2018).  Thinking Clearly About Correlations and Causation: Graphical Causal Models for Observational Data.  Advances in Methods and Practices in Psychological Science, 1(1), 27–42.
  * Rosenbaum (1995) Rosenbaum, P. R. (1995).  Discussion of Causal Diagrams for Empirical Research by J. Pearl.  Biometrika, 82(4), 698–699.
  * Rosenbaum (2010) Rosenbaum, P. R. (2010).  Design of Observational Studies.  Springer Series in Statistics. Springer.
  * Rosenbaum & Rubin (1983) Rosenbaum, P. R., & Rubin, D. B. (1983).  The central role of the propensity score in observational studies for causal effects.  Biometrika, 70(1), 41–55.
  * Rubin (2005) Rubin, D. B. (2005).  Causal Inference Using Potential Outcomes: Design, Modeling, Decisions.  Journal of the American Statistical Association, 100(469), 322–331.
  * Rubin (2007) Rubin, D. B. (2007).  The design versus the analysis of observational studies for causal effects: Parallels with the design of randomized trials.  Statistics in Medicine, 26(1), 20–36.
  * Rubin (2008) Rubin, D. B. (2008).  For objective causal inference, design trumps analysis.  The Annals of Applied Statistics, 2(3), 808–840.
  * Samii (2016) Samii, C. (2016).  Causal Empiricism in Quantitative Research.  The Journal of Politics, 78(3), 941–955.
  * Sekhon (2009) Sekhon, J. S. (2009).  Opiates for the Matches: Matching Methods for Causal Inference.  Annual Review of Political Science, 12(1), 487–508.
  * Shalizi (2021) Shalizi, C. R. (2021).  Advanced Data Analysis from an Elementary Point of View.  (p. 861).
  * Sharkey et al. (2017) Sharkey, P., Torrats-Espinosa, G., & Takyar, D. (2017).  Community and the Crime Decline: The Causal Effect of Local Nonprofits on Violent Crime.  American Sociological Review, 82(6), 1214–1240.
  * Steiner et al. (2017) Steiner, P. M., Kim, Y., Hall, C. E., & Su, D. (2017).  Graphical Models for Quasi-experimental Designs.  Sociological Methods & Research, 46(2), 155–188.
  * VanderWeele & Arah (2011) VanderWeele, T. J., & Arah, O. A. (2011).  Bias Formulas for Sensitivity Analysis of Unmeasured Confounding for General Outcomes, Treatments, and Confounders.  Epidemiology, 22(1), 42–52.
  * Westreich (2019) Westreich, D. (2019).  Epidemiology by Design.  New York, NY: Oxford University Press.
  * Zhou (2019) Zhou, X. (2019).  Equalization or Selection? Reassessing the “Meritocratic Power” of a College Degree in Intergenerational Income Mobility.  American Sociological Review, 84(3), 459–485.


# Spectroscopic evidence of mixed angular momentum symmetry in non-
centrosymmetric Ru7B3

Soumya Datta1, Aastha Vasdev1, Ranjani Ramachandran1, Soumyadip Halder1, Kapil
Motla2, Anshu Kataria2, Arushi2, Rajeswari Roy Chowdhury2, Ravi Prakash
Singh2, and Goutam Sheet1<EMAIL_ADDRESS>1Department of Physical
Sciences, Indian Institute of Science Education and Research Mohali, Sector
81, S. A. S. Nagar, Manauli, PO 140306, India  2Department of Physics, Indian
Institute of Science Education and Research Bhopal, Bhopal 462066, India

###### Abstract

Superconducting crystals with lack of inversion symmetry can potentially host
unconventional pairing. However, till date, no direct conclusive experimental
evidence of such unconventional order parameters in non-centrosymmetric
superconductors has been reported. In this paper, through direct measurement
of the superconducting energy gap by scanning tunnelling spectroscopy, we
report the existence of both $s$-wave (singlet) and $p$-wave (triplet) pairing
symmetries in non-centrosymmetric Ru7B3. Our temperature and magnetic field
dependent studies also indicate that the relative amplitudes of the singlet
and triplet components of the order parameter change differently with
temperature.

In the BCS theory, it is assumed that the attractive interaction that leads to
superconductivity is isotropic in momentum space[1]. Consequently, the
superconducting energy gaps of BCS superconductors show $s$-wave (orbital
angular momentum, $l$=0) symmetry. In certain superconducting systems, the
energy gap may become anisotropic in the momentum space, and show higher
angular momentum symmetries like $p$-wave ($l$=1)[2, 3, 4], $d$-wave
($l$=2)[5, 6, 7] etc. In certain other systems, existence of mixed angular
momentum symmetry, where symmetries represented by different $l$ are mixed,
have also been possible[8, 9, 10, 11]. The physics of such non-$s$-wave
superconductors are not understood within the BCS formalism. If the crystal
structure of a superconductor lacks a centre of inversion symmetry, parity is
not a good quantum number. In such a system an antisymetric spin-orbit
coupling (ASOC) can exist. ASOC can in-principle remove the spin degeneracy of
the Bloch states with same $k$ (crystal momentum), but opposite spins. In
presence of ASOC, the orbital angular momentum and spin angular momentum do
not remain good quantum numbers any longer. Here, Pauli’s exclusion principle
cannot restrict the symmetry of the Cooper pairs to be either purely even-
parity singlet or odd-parity triplet. Therefore, a complex mixed angular
momentum state becomes a possibility in a non-centrosymmetric
superconductor[12]. The unconventionality associated with such complex angular
momentum symmetry of the superconducting order parameters might make the non-
centrosymmetric superconductors (NCS) exhibit unusual behaviour in their
electro-magnetic properties compared to the purely $s$-wave superconductors.
For example, they may display unusually high Pauli limiting fields[13],
helical vortex states[15] and even topologically protected states[16]. Owing
to these, since the discovery of the first non-centrosymmetric superconductor
CePt3Si[13, 14], the study of such superconductors gained significant
attention of the condensed matter physics community[17, 18, 19].

Despite a number of theoretical predictions of the possibility of the exotic
superconducting phases in non-centrosymmetric superconductors as discussed
above, there has been no clear spectroscopic evidence of unconventionality in
such superconductors studied till date. In this paper, we report our ultra-low
temperature scanning tunneling microscopy and spectroscopy results on a non-
centrosymmetric superconductor Ru7B3. Ru7B3 belongs to the space group
$P6_{3}mc$ and the cyclic crystallographic class $C_{6v}$[20]. Matthias $et$
$al.$ had first reported superconductivity in Ru7B3 in 1961[21]. However,
owing to its low critical temperature, the system did not find much interest
among the superconductivity community. Almost three decades later, the absence
of the inversion symmetry in its crystal structure was highlighted by
Morniroli $et$ $al.$[22]. In various transport and thermodynamic measurements
in the past[23, 24, 25], it was seen that $\Delta C_{e}/(\gamma_{n}T_{c})$ and
2$\Delta_{0}/(k_{B}T_{c}$) in Ru7B3 were approximately 1.4 and 3.3
respectively indicating a weak-coupling superconducting state. These
measurements also indirectly indicated that a predominant fully gapped
$s$-wave order parameter could describe the superconducting state of Ru7B3
well. However, as we note, certain special features of the data presented in
[23, 24] were ignored while making a claim for absence of unconventional
superconductivity in Ru7B3. The most intriguing among such special features
was a kink in the field dependent $\rho$-$T$ data[24], beyond which the
superconducting transition curves split into two parts. The two parts
exhibited significantly different sensitivity to the applied magnetic field
and led to two dramatically different field scales for the upper critical
fields ($H_{c2}\sim$ 1.1 T and 5 T respectively). In this context, Fang $et$
$al.$ discussed the possibility of a mixed angular momentum symmetry of the
superconducting order parameter in Ru7B3. In addition, though it was ignored
by the authors, a possible signature of unconventional pairing was also
present in the specific heat data as presented in [23]. More recently, Cameron
$et$ $al.$ performed small-angle neutron scattering [26] on Ru7B3 and reported
that the orientation of the vortex lattice in Ru7B3 strongly depends on the
history of the applied magnetic field thereby indicating the possibility of a
broken time-reversal symmetry in the order parameter. In order to probe the
true order parameter symmetry of Ru7B3, we carried out detailed and direct
temperature and magnetic field dependent scanning tunneling spectroscopy
measurements on Ru7B3. The analysis of such data reveals spectroscopic
signature of an order parameter with mixed angular momentum symmetry.

The single crystals used for our measurements showed a bulk superconducting
transition at 2.6 K (Figure 1(b)). The scanning tunnelling microscopy (STM)
and spectroscopy (STS) experiments were performed in a Unisoku system with RHK
R9 controller, inside an ultra-high vacuum (UHV) cryostat at $\sim$ $10^{-10}$
mbar pressure. The lowest temperature down to which the measurements were
performed was 300 mK. The STM is also equipped with a superconducting solenoid
capable of producing a magnetic field up to 11 T. Since the single crystals
could not be cleaved using the standard cleaving technique (optimized for
layered materials only), we cleaned the surface by reversed sputtering for 30
minutes with Argon (Ar) ion in-situ inside an integrated UHV preparation
chamber. Following that, we immediately transferred the sample to the scanning
stage at low temperature. The Tungsten (W) tip which was prepared outside by
electrochemical etching was also cleaned in UHV by bombarding it with a high-
energy electron-beam. This process helped us probe the pristine surface of
Ru7B3. In the inset of Figure 1(b) we show an STM topographic image showing
the distinctly visible crystallites with average grain size $\sim$ 3 nm.

In Figure 1(a) we show six representative tunneling spectra ($S_{1}$ \-
$S_{6}$) captured at randomly chosen points on the surface of Ru7B3 at $\sim$
310 mK. A visual inspection reveals that based on their overall shapes, the
spectra can be distinctively divided into two categories. The first type
($S_{1}$, $S_{2}$ and $S_{3}$) shows coherence peaks around $\pm$ 0.30 mV and
shallow ‘V’-shaped dip in between the peaks. The spectra ($S_{4}$, $S_{5}$ and
$S_{6}$) of second type show coherence peaks at $\pm$ 0.47 mV and they have a
higher curvature and depth ($dI/dV$ $\sim$ 0 at $V$=0) below that. We have
also analysed these spectra within a single band ‘$s$-wave’ model[1] using
Dyne’s formula[27]: $N_{s}(E)\propto
Re\left(\frac{(E-i\Gamma)}{\sqrt{(E-i\Gamma)^{2}-\Delta^{2}}}\right)$. The
tunnelling current is given by
$I(V)\propto\int_{-\infty}^{+\infty}N_{s}(E)N_{n}(E-eV)[f(E)-f(E-eV)]dE$.
Here, $N_{s}(E)$ and $N_{n}(E)$ are the density of states (DOS) of the
superconducting sample and the normal metallic tip respectively, while $f(E)$
is the Fermi-Dirac distribution function. $\Gamma$ is the Dyne’s parameter
that takes care of broadening of DOS. The theoretical plots thus generated are
shown as black lines on the experimental data points in the Figure. It is seen
that the spectra $S_{1}$, $S_{2}$ and $S_{3}$ give $\Delta$ of the order of
0.31 meV, 0.29 meV and 0.24 meV respectively while spectra $S_{4}$, $S_{5}$
and $S_{6}$ provide the values 0.48 meV, 0.42 meV and 0.49 meV respectively
for the same analysis. It is also noted that, while the first group of spectra
($S_{1}$, $S_{2}$ and $S_{3}$) shows reasonably good fitting with the single
gap ‘$s$-wave’ model (albeit with large $\Gamma$), the second group ($S_{4}$,
$S_{5}$ and $S_{6}$) exhibits a significant departure from that.

The above-mentioned discrepancy between the experimental spectra and the
spectra generated theoretically within a single-band ‘$s$-wave’ model,
prompted us to consider other possible symmetries of the order parameter. To
perform such an analysis, we modified Dyne’s equation by introducing a more
general expression[28] of $\Delta(\theta)$ than an isotropic $\Delta$. The
modified Dyne’s equation reads as $N_{s}(E,\theta)\propto
Re\left(\frac{(E-i\Gamma)}{\sqrt{(E-i\Gamma)^{2}-(\Delta^{\prime}Cos(n\theta))^{2}}}\right)$.
Here, $\theta$ is the polar angle (w.r.t. $(001)$) and the integer $n$ can be
0, 1 or 2 for $s$, $p$ and $d$ wave symmetries respectively. The expression
for tunnelling current is also modified to
$I(V)\propto\int_{-\infty}^{+\infty}\int_{0}^{2\pi}N_{s}(E,\theta)N_{n}(E-eV)[f(E)-f(E-eV)]d\theta
dE$. In Figure 1(c) we show the experimental spectrum $S_{4}$ along with
theoretical plots considering isotropic ‘$s$-wave’ $\Delta$ (red line) and
anisotropic ‘$p$-wave’ $\Delta$ (blue line). It is clear that the spectrum,
especially the ‘V’-shaped part of that between the coherence peaks, is better
described by the ‘$p$-wave’ symmetry. It is also interesting to note that, the
extracted value of $\Delta$ (0.47 meV) for such fit does not differ much with
the same from the best ‘$s$-wave’ fit (0.48 meV). Such ‘V’-shaped spectra are
often seen for superconductors with possible unconventional symmetries and are
well described by the Tanaka-Kashiwaya model[28] we used here. For example,
from tunnelling spectroscopic study on SmFeAsO0.85, Millo $et$ $al.$[29]
associated such a shape with an unconventional order parameter. To note, it
was also reported there that some of the STM spectra could also be fitted well
within pure ‘$s$-wave’ model but with significantly smaller $\Delta$ and
relatively higher $\Gamma$ – a situation similar to our group-I spectra.

Now we focus on the spectrum $S_{1}$, which belongs to group I and fits
reasonably well with single ‘$s$-wave’ gap (and with relatively large
$\Gamma$). A closer inspection, however, reveals that there is a small
discrepancy between the experimental data and the ‘$s$-wave’ model spectrum.
We investigated the evolution of this discrepancy with temperature.
Temperature dependence of $\Delta$ approximately followed the BCS prediction
(Figure 2(c)) with $\Delta_{0}$ = 0.3 meV. The broadening parameter $\Gamma$
did not change much within this range. The departure of the experimental
spectrum from the $s$-wave model rapidly decreased with increasing temperature
and at around 750 mK the discrepancy almost disappeared (Figure 2(a)). To
illustrate this effect clearly, we show two spectra, one at 338 mK and another
at 961 mK along with their theoretical ($s$-wave) fits in Figure 2(b). This
observation indicates the possibility of a mixed angular momentum symmetry in
the order parameter where the amplitudes corresponding to different $l$ vary
differently with temperature. This also explains why Fang $et$ $al.$ did not
find any signature of unconventionality in their lower critical field
($H_{c1}$) studies[24] down to 1.2 K, which is well above the temperature
window where we could notice deviation from $s$-wave behaviour in our data. We
have also performed magnetic field dependence of another spectrum ($S_{3}$)
from the same group I (Figure 3(a)). The key observation is that while the
spectra deviate appreciably from ‘$s$-wave’ theoretical curves at low fields,
above 10 kG the spectra resembled far more closely to the ‘$s$-wave’
predictions. For demonstrating this effect more clearly, we show the spectra
and respective ‘$s$-wave’ fits at zero field and at 16 kG in Figure 3(b). The
evolution of the extracted gap ($\Delta$) and broadening parameter ($\Gamma$)
with the applied magnetic field are shown in Figure 3(c).

In Figure 4(a) we demonstrated the detailed temperature dependent study of
spectra $S_{6}$ which belong to group II. At $\sim$ 740 mK a small zero-bias
conductance peak (ZBCP) appeared. It became more and more pronounced up to
$\sim$ 1370 mK and then started fading until it, along with other prominent
spectral features, gradually disappeared at 1.9 K. To note, emergence of such
a ZBCP can be attributed to a ‘$p$-wave’ component in the order parameter
where the interface normal and the lobe-direction of the ‘$p$-wave’ maintain
an acute angle between them[30, 31]. Since the surface of our Ru7B3 has
crystallites with random orientations and the tip is engaged randomly at
different points, this condition can naturally be satisfied some times.

Based on the discussion above, if Ru7B3 has a mixed angular momentum symmetry
in its order parameter, then we would expect the unconventional component to
be present in group-I spectra too. To understand this aspect in details, we
used an ‘$s+p$-wave’ model to analyse the spectra $S_{1}$. In this model, the
effective gap is given by $\Delta_{s+p}$ = $\Delta_{s}$ \+
$\Delta_{p}Cos\theta$. In Figure 4(b) we present the spectrum $S_{1}$ (black
circles) at 338 mK with numerically generated spectra considering ‘$s+p$-wave’
symmetry (blue line). Pure ‘$s$-wave’ fit (red line) is also shown for
comparison. It is clear that the mixed angular momentum symmetry provides a
better description of the data. Temperature evolutions for the amplitudes of
the two components $\Delta_{s}$ and $\Delta_{p}$ extracted from the above
‘$s+p$-wave’ fittings for the whole spectra $S_{1}$ are presented in Figure
4(c). Not as a surprise, while the convensional $\Delta_{s}$ follows a smooth
BCS like dependence upto 2 K, the smaller $\Delta_{p}$ sharply drops and goes
beyond our measurement resolution at 0.9 K.

It should be noted here that the superconducting gap is the manifestation of a
phase-coherent macroscopic condensate. Therefore, the measured $\Delta$ should
ideally be unique irrespective of the measurement techniques. Since the system
is metallic in nature, any role of special surface states is ruled out. The
$T_{c}$ and the $H_{c}$ that we measure for all our spectra match well
irrespective of whether they fall under Group I or II. Therefore, it can be
concluded that, all the spectra falling under two distinct groups differ with
each other based on which component of the order parameter contributes
predominantly for a particular crystallite that the measurement is performed
on.

In summary, we performed scanning tunnelling spectroscopy on single-crystal
Ru7B3 and recorded several spectra which can be broadly categorised into two
groups, group-I and II. Group-I consists of spectra which are shallow in
shape. They show overall good agreement with single gap ‘$s$-wave’ symmetry
except at very low temperatures where they deviate from such pure $s$-wave
model. Smaller superconducting gap and larger broadening parameter are
characteristics of these spectra. Group-II spectra are broader in shape and
show sharper coherence peaks. They significantly deviate from the predictions
of ‘$s$-wave’ model but a theoretical model with ‘$p$-wave’ symmetry shows
better agreement. The superconducting gap is larger, and the broadening
parameter is very small for such spectra compared to those belonging to the
first group. The temperature dependences of the spectra belonging to both the
groups indicate the presence of a mixed ‘$s+p$-wave’ symmetry in the order
parameter where the two components have different temperature dependence.

We thank Tanmoy Das for his useful comments. R.P.S. acknowledges the financial
support from the Science and Engineering Research Board (SERB)-Core Research
Grant (grant No. CRG/2019/001028). G.S. would like to acknowledge financial
support from the Swarnajayanti fellowship awarded by the Department of Science
and Technology (DST), Govt. of India (grant No. DST/SJF/PSA-01/2015-16).

## References

  * [1] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
  * [2] H. R. Ott, H. Rudigier, T. M. Rice, K. Ueda, Z. Fisk, and J. L. Smith, Phys. Rev. Lett. 52, 1915 (1984).
  * [3] K. Ueda and T. M. Rice, Phys. Rev. B 31, 7114 (1985).
  * [4] Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, K. Ishida, J. Phys. Soc. Jpn. 81, 011009 (2012).
  * [5] D. A. Wollman, D. J. Van Harlingen, W. C. Lee, D. M. Ginsberg, and A. J. Leggett, Phys. Rev. Lett. 71, 2134 (1993).
  * [6] C. C. Tsuei, J. R. Kirtley, C. C. Chi, Lock See Yu-Jahnes, A. Gupta, T. Shaw, J. Z. Sun, and M. B. Ketchen, Phys. Rev. Lett. 73, 593 (1994).
  * [7] C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, 969 (2000).
  * [8] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schäfer, Phys. Rev. Lett. 43, 1892 (1979).
  * [9] T. A. Tokuyasu, D. W. Hess, and J. A. Sauls, Phys. Rev. B 41, 8891 (1990).
  * [10] K. A. Kouznetsov, A. G. Sun, B. Chen, A. S. Katz, S. R. Bahcall, John Clarke, R. C. Dynes, D. A. Gajewski, S. H. Han, M. B. Maple, J. Giapintzakis, J.-T. Kim, and D. M. Ginsberg, Phys. Rev. Lett. 79, 3050 (1997).
  * [11] J. F. Annett, Adv. Phys., 39 83 (1990).
  * [12] P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004).
  * [13] E. Bauer, G. Hilscher, H. Michor, Ch. Paul, E. W. Scheidt, A. Gribanov, Yu. Seropegin, H. Noël, M. Sigrist, and P. Rogl, Phys. Rev. Lett. 92, 027003 (2004).
  * [14] K. V. Samokhin, E. S. Zijlstra, and S. K. Bose, Phys. Rev. B 69, 094514 (2004).
  * [15] R. P. Kaur, D. F. Agterberg, and M. Sigrist, Phys. Rev. Lett. 94, 137002 (2005).
  * [16] M. Sato, S. Fujimoto, Phys. Rev. B 79, 094504 (2009).
  * [17] M. Sigrist, D. F. Agterberg, P. A. Frigeri, N. Hayashi, R. P. Kaur, A. Koga, I. Milat, K. Wakabayashi, and Y. Yanase, J. Magn. Magn. Mater. 310, 536 (2007).
  * [18] A.D. Hillier, J. Quintanilla, R. Cywinski, Phys. Rev. Lett. 102, 117007 (2009).
  * [19] E. Bauer, M. Sigrist, Non-Centrosymmetric Superconductors: Introduction and Overview, Berlin, Heidelberg, (2012).
  * [20] B. Aronsson, Acta Chem. Scand. 13, 109 (1959).
  * [21] B. T. Matthias, V. B. Compton, and E. Corenzwit, J. Phys. Chem. Solids 19, 130 (1961).
  * [22] J. P. Morniroli, H. Ayatti, K. M. Knowles, W. M. Stobbs, and M. Gantois, J. Less-Common Met. 155, 215 (1989).
  * [23] N. Kase and J. Akimitsu, J. Phys. Soc. Jpn. 78, 044710 (2009).
  * [24] L. Fang, H. Yang, X. Zhu, G. Mu, Z.-S. Wang, L. Shan, C. Ren, and H.-H. Wen, Phys. Rev. B 79, 144509 (2009).
  * [25] R. Singh, N. Parzyk, M. Lees, D. Paul, and G. Balakrishnan, J. Cryst. Growth 395, 22 (2014).
  * [26] A. S. Cameron, Y. S. Yerin, Y. V. Tymoshenko, P. Y. Portnichenko, A. S. Sukhanov, M. Ciomaga Hatnean, D. Mc K. Paul, G. Balakrishnan, R. Cubitt, A. Heinemann, and D. S. Inosov, Phys. Rev. B 100, 024518 (2019).
  * [27] R. C. Dynes, V. Narayanamurti, and J. P. Garno, Phys. Rev. Lett. 41, 1509 (1978).
  * [28] Y. Tanaka and S. Kashiwaya, J. Phys. Soc. Jpn. 74, 3451 (1995)
  * [29] O. Millo, I. Asulin, O. Yuli, I. Felner, Z.-A. Ren, X.-L. Shen, G.-C. Che, and Z.-X. Zhao, Phys. Rev. B 78, 092505 (2008)
  * [30] M. Yamashiro, Y. Tanaka, N. Yoshida, and S. Kashiwaya, J. Phys. Soc. Jpn. 68, 2019 (1999).
  * [31] F. Laube, G. Goll, M. Eschrig, M. Fogelström, and R. Werner, Phys. Rev. B 69, 014516 (2004).

Figure 1: (a) Six representative tunnelling spectra ($S_{1}$ \- $S_{6}$) plots
(colour points) along with corresponding numerically generated spectra under
single gap $s$-wave model (black lines). The extracted fitting parameters
$\Delta$ and $\Gamma$ are shown also for each spectrum. (c) Spectrum $S_{4}$
with both best single gap ‘$s$-wave’ and ‘$p$-wave’ fit alongwith
corresponding extracted parameters. The temperature ($T$) $\sim$ 310 mK for
all spectra. (b) Bulk magnetization ($M$) data in both zero field cool warming
(ZFCW) and field cool cooling (FCC) condition with 10 G magnetic field.
$inset$: STM topographic image of the sample. Figure 2: (a) Temperature (T)
dependence of tunnelling conductance spectra $S_{1}$ (colour lines) with
theoretical fits (black lines) in absence of any magnetic field. (b) Spectra
$S_{1}$ at 340 mK and also at 960 mK along with corresponding fitting
parameters, where better fit at higher temeperature is visible. (c) Evolution
of $\Delta$ and $\Gamma$ with temperature, extracted from the plot (a) along
with an ideal BCS trend of $\Delta$ for comparison. Figure 3: (a) Magnetic
field ($H$) dependence of tunnelling conductance spectra $S_{3}$ (colour
lines) with theoretical fits (black lines) all measured at T $\sim$ 310 mK.
(b) Spectra $S_{3}$ in the environment of $H$ = 0 and $H$ = 16 kG field, along
with corresponding fitting parameters. $H\parallel c$-axis of the crystal and
a better fit at higher field is visible. (c) Evolution of $\Delta$ and
$\Gamma$ with magnetic field, extracted from the plot (a). Figure 4: (a)
Temperature (T) dependence of tunnelling conductance spectra $S_{6}$
incorporating gradual appearance and disappearance of the peak like feature.
(b) Spectrum $S_{1}$ along with best pure $s$-wave (red line) and mixed
$s+p$-wave (blue line) fits with corresponding extracted parameters $\Delta$
and $\Gamma$. (c) Evolution of $\Delta_{s}$ and $\Delta_{p}$ with temperature,
extracted from the $s+p$-wave fits of spectra $S_{1}$. Ideal BCS trends of
$\Delta$ are also shown for comparison.