# UoB at SemEval-2020 Task 1: Automatic Identification of Novel Word Senses

Eleri Sarsfield
School of Computer Science
University of Birmingham
United Kingdom
<EMAIL_ADDRESS>
&Harish Tayyar Madabushi
School of Computer Science
University of Birmingham
United Kingdom
<EMAIL_ADDRESS>

###### Abstract

Much as the social landscape in which languages are spoken shifts, language
too evolves to suit the needs of its users. Lexical semantic change analysis
is a burgeoning field of semantic analysis which aims to trace changes in the
meanings of words over time. This paper presents an approach to lexical
semantic change detection based on Bayesian word sense induction suitable for
novel word sense identification. This approach is used for a submission to
SemEval-2020 Task 1, which shows the approach to be capable of the SemEval
task. The same approach is also applied to a corpus gleaned from 15 years of
Twitter data, the results of which are then used to identify words which may
be instances of slang.

##  1 Introduction

00footnotetext:  Accepted for publication at SemEval 2020
This work is licensed under a Creative Commons Attribution 4.0 International
Licence. Licence details: http://creativecommons.org/licenses/by/4.0/.

Automatic lexical semantic change detection is a field of semantic analysis
which aims to discern how the meanings of words change over time. As interest
in the field has increased, a variety of different procedures, languages and
corpora have been used, which leads to difficulty when attempting to compare
different sets of results. SemEval-2020 Task 1, “Unsupervised Lexical Semantic
Change Detection” [Schlechtweg et al. (2020], is a task aimed at providing a
single unified framework with which to compare approaches using a standardised
dataset, in order to address the difficulty in attempting to compare different
sets of results which arises due to the variety in procedures, languages and
corpora which have previously been used. The task involves determining whether
a set of target words have changed meaning in two corpora, each of which
corresponds to a different time period. Corpora are provided for German,
English, Latin and Swedish. The task consists of two subtasks, one involving a
binary classification of target words into words which have or have not
changed meaning, the other involving a ranking of words according to degree of
change.

This paper describes our submission to SemEval-2020 Task 1111Source code and
data are published at https://github.com/elerisarsfield/semeval. To produce
this submission, we use an approach based on the Hierarchical Dirichlet
Process (HDP) [Teh et al. (2004], an extension of Latent Dirichlet Allocation
(LDA) which allows the number of senses to be unbounded. The next section
presents works related to methods for identifying novel word senses, which is
followed by a description of our system in Section 3 and the results of the
application of the system in Section 4. Section 5 provides an overview of our
conclusions and possible directions of future work. We also use our method on
a corpus constructed from data from Twitter in order to explore the
possibility of detecting semantic change over a shorter period of time, with a
focus on where that detection can be used for identifying slang.

##  2 Related Work

Approaches to lexical semantic change detection collect co-occurrence
information on words, represented in most approaches by pointwise mutual
information scores [Tahmasebi et al. (2018]. This information may be used as
the basis of an approach, as is used by ?) and ?) to identify semantic change
over time by using context vectors. Context vectors can also be used as part
of a different representation. ?) use a PPMI matrix for each time period to
learn dynamic embeddings, an approach to identifying lexical change which
reduces the need to align vectors that arises when using the static embeddings
which are traditionally used. ?) also use dynamic embeddings, presenting a
probabilistic version of word2vec ?), an architecture for static embeddings.
Embeddings are also used by ?), who use domain-specific embeddings to detect
domain shifts. A survey of the use of embeddings for semantic shifts detection
is given in ?).

Context vector and embeddings approaches to identifying change of meaning do
not have the ability to recover the word senses. ?) give an approach to the
problem of word sense induction which is based on the Latent Dirichlet
Allocation (LDA) model [Blei et al. (2003] of text generation. The LDA model
is extended from parametric to nonparametric space by ?) with an infinite
vocabulary. ?) instead use a Hierarchical Dirichlet Process (HDP) model [Teh
et al. (2004] for the task of word sense induction. A model based on the HDP
is similarly used by ?), for extending the flexibility of word sense induction
models in order to adapt to varying degrees of polysemy. An HDP is also used
by ?), who apply word sense induction to dictionaries and newspapers. A
similar approach is taken by ?), in the context of updating dictionaries. ?)
use a Bayesian model for tracking gradual sense changes over time.

Graph-based models to finding lexical semantic change are proposed by ?),
tracking language evolution by clustering word senses over various time
periods, an approach echoed by ?), studying word sense change in digitised
books from 1520 to 2008. The most comprehensive survey of the field of lexical
semantic change detection is presented by ?).

##  3 System Overview

The data from SemEval-2020 Task 1 consists of two corpora for each language,
each consisting of a number of short contexts of words. Though data are
provided in a number of languages, only the data for English was used. Of each
of the two corpora, one corpus corresponds to a reference corpus and the other
to a focus corpus. The reference corpus is taken to represent standard usage,
and the focus corpus contains newer texts. Each corpus is partitioned into
short pseudo-documents, which may be treated as documents. ‘Pseudo-document’
and ‘document’ will hereafter be used interchangeably. The creation of these
pseudo-documents is under the assumption that each corpus takes the form of
short contexts of a small number of sentences of text. All stopwords and low
frequency terms are removed.

Notation | Description
---|---
$n_{jt}$ | Number of words in document $j$ at table $t$
$n_{jt}^{-ji}$ | Number of words in document $j$ at table $t$ except $x_{ji}$
$m_{\cdot k}$ | Number of tables with topic $k$ in the corpus
$m_{\cdot\cdot}$ | Number of tables in the corpus
$f_{k}^{-x_{ji}}(x_{ji})$ | Conditional density of $x_{ji}$ under mixture component $k$ given all items except $x_{ji}$
$f_{k}^{-\bm{x}_{jt}}(\bm{x}_{jt})$ | Conditional density of $\bm{x}_{jt}$ given all items associated with $k$ except $\bm{x}_{jt}$
Table 1: Notation used in the Chinese Restaurant Franchise

A Word Sense Induction (WSI) model based on a Hierarchical Dirichlet Process
(HDP), a clustering approach which allows clusters to be shared among the
groups [Teh et al. (2004], is used to model the senses for each word. Notation
is given in Table 1. An HDP is selected over a Latent Dirichlet Allocation
(LDA) model as used in ?) because the HDP allows the number of latent factors
in the HDP to grow with the data.

Figure 1: A depiction of a Chinese Restaurant Franchise [Teh et al. (2004].
Customers $\theta_{i}$ sit at tables $\psi_{t}$ which each order menu items
$\phi_{k}$

An initial random set of senses is induced modelled after the generative
process of the HDP, which corresponds to a partition of words inside
documents, a process based on the Chinese Restaurant Franchise (CRF)
representation of a two level HDP [Teh et al. (2004], which partitions
customers at the group level and dishes at the top level. Each document $j$
may be considered a group. Its document level Chinese Restaurant Process (CRP)
generates a table index $t_{ji}$ for each observation $i$ according to

$p(t_{ji}=t|t_{j1},\cdots,t_{j,i-1},\alpha)\propto\begin{cases}n_{jt}&\text{if
}t\text{ previously used}\\\ \alpha&\text{if }t=t^{new}\end{cases}$ (1)

where $\alpha$ is a concentration parameter. This partitions the document $j$
into tables.

After all words are assigned to a table, the topic index $k_{jt}$ of all
tables are generated by the corpus-level CRP. Each table is assigned a topic
index according to

$p(k_{jt}|k_{11},\cdots,k_{j,t-1},\gamma)\propto\begin{cases}m_{\cdot
k}&\text{if }k\text{ previously used}\\\ \gamma&\text{if
}k=k^{new}\end{cases}$ (2)

where $\gamma$ is a concentration parameter for the base distribution. This
assigns tables to topics, which completes the initial partition. Each
observation $x_{ji}$ is associated with a table index $t_{ji}$, which is in
turn associated with a topic index $k_{jt}$. This topic index links the table
to one of the corpus topics $\phi_{k}$. This partition is updated using the
sampling scheme outlined in ?), which we summarise shortly.

Each word in each document is assigned to a per-document table $t_{ji}$, with
each of these tables associated with a factor $k_{jt}$. The conditional
distribution of $t_{ji}$ is calculated by combining the conditional prior
distribution for $t_{ji}$ with the likelihood of generating $x_{ji}$. The
prior probability of $t_{ji}$ taking a previously used value is proportional
to $n_{jt}^{-ji}$. The probability it takes on a new value is proportional to
the parameter $\alpha$. The likelihood due to $x_{ji}$ of $t_{ji}$ for a
previously used $t$ is $f_{k}^{-x_{ji}}(x_{ji})$. For $t_{ji}=t^{new}$ the
likelihood is

$p(x_{ji}|\bm{t}^{-ji},t_{ji}=t^{\text{new}},\bm{k})=\sum_{k=1}^{K}\frac{m_{\cdot
k}}{m_{\cdot\cdot}+\gamma}f_{k}^{-x_{ji}}(x_{ji})+\frac{\gamma}{m_{\cdot\cdot}+\gamma}f_{k^{\text{new}}}^{-x_{ji}}(x_{ji})$
(3)

where $f_{k^{\text{new}}}^{-x_{ji}}(x_{ji})=\int f(x_{ji}|\phi)h(\phi)d\phi$
is the prior density of $x_{ji}$. From this, the conditional distribution of
$t_{ji}$ can be formed as

$p(t_{ji}=t|\bm{t}^{-ji},\bm{k})\propto\begin{cases}n_{jt}^{-ji}f_{k_{jt}}^{-x_{ji}}(x_{ji})&\text{if
}t\text{ previously used}\\\ \alpha
p(x_{ji}|\bm{t}^{-ji},t_{ji}=t^{\text{new}},\bm{k})&\text{if
}t=t^{\text{new}}\end{cases}$ (4)

This provides the distribution according to which the tables for words are
sampled. The PPMI values in the co-occurrences matrix are used for estimation.
If the sampled value is $t^{\text{new}}$, then a new table is created for the
restaurant and the word assigned to it. A dish must then be selected for this
table. A sample for $k_{jt^{\text{new}}}$ is obtained according to

$p(k_{jt^{\text{new}}}=k|\bm{t},\bm{k}^{-jt^{\text{new}}})\propto\begin{cases}m_{\cdot
k}f_{k}^{-x_{ji}}(x_{ji})&\text{if }k\text{ previously used}\\\ \gamma
f_{k^{\text{new}}}^{-x_{ji}}(x_{ji})&\text{if }k=k^{\text{new}}\end{cases}$
(5)

This gives an assignment of each word to a cluster inside a document, and of
each cluster to a corpus-level topic. Senses are formed from each topic by
considering the distribution of assigned words. These topics represent the
senses across both corpora, inferred on pooled instances of each word. Since
the topics are jointly modelled, discovered topics are applicable to both
corpora, which means that there is no need to reconcile these senses [Cook et
al. (2014]. Words with multiple senses are represented by their position in
multiple topics. Once sampling has concluded, the word distributions over
topics are calculated. Word distributions are calculated based on the topics
assigned to each observed word $x_{ji}$ in each document to determine a
distribution over topics separately for each word for the reference and focus
instances. From this, the Jensen-Shannon distance for the target words between
the distribution in the reference and focus corpus is calculated.

Additionally, a method for the application to the Twitter corpus is used based
on the NoveltyDiff method in ?), a description of which follows. The
distribution of words over senses is categorised according to which instances
originated in the reference corpus and which in the focus corpus. For each
sense $s$, the novelty score is calculated as

$\text{Novelty}_{\text{Diff}}(s)=p_{f}(s)-p_{r}(s)$ (6)

where $p_{f}(s)$ and $p_{r}(s)$ are the proportion of usages of a given word
corresponding to sense $s$ in the focus and reference corpus, respectively.
The score for each word is the maximum for any of its induced senses. Once the
novelty scores have been calculated for all words, the words with novel senses
can be identified as those with the highest scores of NoveltyDiff. This will
be used when applying the model to Twitter data rather than the SemEval task
in order to identify which words have gained senses, a narrower focus which is
useful in the applied context of identifying slang.

##  4 Results

The model’s performance on SemEval-2020 Task 1 is presented in Table 2,
alongside results from the baselines for each subtask. The model was only run
against the English data for each subtask, using the sample files for all
other languages to form a complete submission. To reflect this, we record both
the overall score and the score based on performance on the English component
in Table 2. The frequency difference baseline is the absolute difference in
normalised target word frequencies in the corpora. The count vector baseline
is the cosine difference between vector representations of words in the two
corpora. The values for these baselines are provided by the task
organisers222https://competitions.codalab.org/competitions/20948\\#results.
The model was run for 1 iteration. The window size was set to 2, the floor for
determining low frequency terms was set to 1, and the concentration parameters
$\alpha$ and $\gamma$ were both set to 1. To produce the submission, the
Jensen-Shannon score for each target is calculated. This provides the
submission for Subtask 2. For Subtask 1, lemmas are determined to have changed
meaning if their Jensen-Shannon distance is above a threshold value of 0.6.

Subtask | Score (Overall) | Score (English) | Frequency Difference | Count Vector | Rank
---|---|---|---|---|---
1 | 0.526 | 0.568 | 0.432 | 0.595 | 19
2 | 0.100 | 0.105 | -0.217 | 0.022 | 18
Table 2: Results for SemEval 2020 Task 1 for English data. Baselines scores
are taken from the English component only.

The results from the SemEval competition (see Table 2 for complete results),
for which we only produced a submission for the English component, led to a
placement, when considering performance on the English data only, of 7th of
9th for the classification subtask and 16th of 21st for the ranking subtask,
and were more accurate than both of the baselines for the ranking subtask, and
for one of the baselines for the classification subtask, indicating the model
works as expected to a reasonable degree of accuracy. The lacklustre
performance on the first subtask as compared with the second may have been
influenced by the choice of threshold parameter, which was never altered and
chosen with little empirical evidence, which resulted in a threshold parameter
which was too low and as such was unable to appropriately identify instances
of meaning change. Experimenting with this value, as well as those of other
hyperparameters, would likely have improved performance.

Word |  Novel Definition
---|---
ill |  cool, tight, sweet
hater |  a person that simply cannot be happy for another person’s success. So rather than be happy they make a point of exposing a flaw in that person.
like |  a meaningless word teenagers insert liberally into both colloquial and formal speech in order to maintain a steady stream of words
roll |  used to describe the effects of Ecstasy
safe |  a cool person;to signify agreement;to signify something is good
checked |  the process by which someone puts another individual in their place verbally or physically either in a joking manner or a serious beatdown.
bars |  (1) sentences in lyrical hiphop songs (2) slang name for a 2mg Xanax tablet
Table 3: Identified words and definitions

In addition to the SemEval competition, the same model was also used to
experiment with data from Twitter with the goal of determining whether the
same procedure can be used for sense change over a shorter timeframe, with a
particular focus on identifying slang through words which had novel senses.
The corpus was constructed from a set of 500,000 tweets, spanning a period of
15 years. Using the approach detailed in Section 3 with the Twitter data
produced a set of 7 slang words. These results are the words identified as
having the most difference in the senses between the two corpora according to
the NoveltyDiff metric. The model was run for 1 iteration against the Twitter
corpora in order to identify the lemmas determined as having most differed
between the older and more recent corpus, and therefore most likely to have
gained a novel sense. The 25 words with the highest value of
$\text{Novelty}_{\text{Diff}}$ were extracted. The online slang dictionary
Urban Dictionary333https://www.urbandictionary.com/ was used to verify whether
or not each of these words had a meaning aside from its standard usage or was
generated erroneously. Where an identified word was a genuine instance of
slang, it is listed alongside this sense as defined by Urban Dictionary in
Table 3. Of the identified words, 7 were able to be verified. The full set of
words which were identified is available on GitHub.

##  5 Conclusions and Future Work

This paper used Bayesian word sense induction methods as the basis for
identifying lexical semantic change, and produced a model for determining
sense differences which was then evaluated on Task 1 in the SemEval-2020
workshop, which confirmed the method is appropriate to the task and works as
expected. The same approach was also used with data from Twitter with the goal
of identifying slang, which produced a set of candidate words. Whilst some of
these were genuine instances of slang, a significant amount were erroneously
identified, or amounted to mere lexical innovation rather than any true
semantic change.

The system was only tested against the English data for each subtask. Future
work would involve extending the system into all of the competition’s
languages.

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††thanks: Corresponding author<EMAIL_ADDRESS>Corresponding author<EMAIL_ADDRESS>
# Nano-scale dynamics of hydrogen in VO2 studied by $\mu$SR

H. Okabe  Muon Science Laboratory and Condensed Matter Research Center,
Institute of Materials Structure Science, High Energy Accelerator Research
Organization (KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan  Institute for
Materials Research, Tohoku University (IMR), 2-1-1 Katahira, Aoba-ku, Sendai
980-8577, Japan  M. Hiraishi  Muon Science Laboratory and Condensed Matter
Research Center, Institute of Materials Structure Science, High Energy
Accelerator Research Organization (KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan
Graduate School of Science and Engineering, Ibaraki University 2-1-1 Bunkyo,
Mito, Ibaraki, 310-8512, Japan  A. Koda  Muon Science Laboratory and Condensed
Matter Research Center, Institute of Materials Structure Science, High Energy
Accelerator Research Organization (KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan
Department of Materials Structure Science, The Graduate University for
Advanced Studies (Sokendai), Tsukuba, Ibaraki 305-0801, Japan  Y. Matsushita
National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0044,
Japan  T. Ohsawa  National Institute for Materials Science (NIMS), Tsukuba,
Ibaraki 305-0044, Japan  N. Ohashi  National Institute for Materials Science
(NIMS), Tsukuba, Ibaraki 305-0044, Japan  R. Kadono  Muon Science Laboratory
and Condensed Matter Research Center, Institute of Materials Structure
Science, High Energy Accelerator Research Organization (KEK-IMSS), Tsukuba,
Ibaraki 305-0801, Japan  Department of Materials Structure Science, The
Graduate University for Advanced Studies (Sokendai), Tsukuba, Ibaraki
305-0801, Japan

###### Abstract

Hydrogen dynamics in the nanoscale region of VO2 was investigated by muon spin
rotation/relaxation ($\mu$SR) technique. Positively charged muon acts as a
light radioisotope of proton and can be used as a probe to explore the inside
of materials from an atomic perspective. By analyzing the muon hopping rate
and the spatial distribution of the 51V nuclear magnetic moments, we have
identified two types of diffusion paths in VO2 (via oxygen-muon bonds or
defects) and the potential of a high diffusion coefficient in the 10-10 cm2/s
range at ambient temperature. Our results provide valuable information for
developing hydrogen-driven VO2–related electronic devices.

##  I Introduction

Vanadium dioxide (VO2) has drawn the interest of numerous researchers due to
its pronounced thermal-driven metal-insulator (M-I) transition at $T_{\rm MI}$
(= 340 K) with an enticing V-V dimerization in the low-temperature phase Morin
. In parallel with the fierce debate on the M-I transition mechanism, much
effort has been attempted to explore its device applications Jero ; Cav .
Since the resistance varies over several orders of magnitude across the M-I
transition and the transition temperature can be tuned over a wide range by
impurity doping, various applications are being explored. One of the
applications is resistive random access memory (ReRAM) Sawa , which has the
potential to realize next-generation nonvolatile memory devices with high-
speed operation, high current density, and high-density recording by using VO2
as a selector Son . Furthermore, ReRAM may apply to artificial neural networks
due to its multi-level, analog memory, and hysteretic current-voltage (I-V)
characteristics Waser .

However, in recent years, Ji et al. pointed out that electrical switching
behavior in VO2-based electric double-layer transistors (EDLT) arises from the
electrochemically driven doping of hydrogen ions Ji . It has been reported
that impurity hydrogen in the ionic liquid used for the EDLT electrode
migrates into VO2 by the electric field and provides carriers, causing a
change in resistance Zhou ; Shyb . In general, hydrogen readily penetrates
metal oxide semiconductors and is a possible source for unintentional carrier
doping Wale1 ; Wale2 . It has also been found that a considerable amount of
hydrogen can reversibly enter and desorb from the VO2 film, significantly
affecting its electrical properties Andre ; Yoon . Hydrogenation of VO2 also
induces structural modification; high hydrogen content leads to orthorhombic
HVO2 where intercalated hydrogen accommodates in an oxygen (O)-channel,
forming typical OH bonds Filin . Therefore, the physical properties of VO2 are
strongly affected by hydrogen, and information on hydrogen dynamics is crucial
for developing these devices.

For hydrogen dynamics in VO2, diffusion coefficients estimated from optical
observations of the metal-insulator domain boundary and from measurements of
transient electronic transport properties during proton intercalation have
been reported Lin ; Mura . Since these reports focus on the micron-order
VO2-HVO2 transition process, a proton-relay hydrogen diffusion mechanism
(Grotthuss mechanism Gro ) with a high hydrogen concentration seems dominant.
In contrast, VO2 in electronic devices is typically a thin film of several
tens of nm, and therefore the hydrogen content is small and may have a
different diffusion mechanism than the above. However, observing the dynamics
of trace amounts of hydrogen in devices is extremely difficult and remains
unexplored.

Here, we report on the dynamics of hydrogen at a dilute concentration in VO2
investigated by muon spin rotation ($\mu$SR) spectroscopy. Positively charged
muon ($\mu^{+}$) is a subatomic particle that acts as a light radioisotope of
proton, simulating the local state of hydrogen in the matter on an atomic
scale while simultaneously probing one’s own evironment Seeg . In addition,
muons are injected directly into the material as a pulsed ion beam generated
by an accelerator (about 103/cm2 per pulse), so there is no need to fabricate
the sample into a thin film or device. The clarification of hydrogen dynamics
in VO2 in the nanoscale region using muons is expected to provide valuable
insights into the development of next-generation electronic devices such as
ReRAM and hydrogen-driven devices.

##  II Experimental Details

We used a powder sample (M180 $\mu$m pass) of VO2 (99.9% purity, Kojundo
Chemical Lab. Co., Ltd.). The crystal structure and phase purity were checked
using the powder X-ray diffractometer (SmartLab, Rigaku Co.) from 300 to 400 K
(see the Supplemental Material, Fig. S1-S3). The structural transition from
the low-temperature orthorhombic phase to the high-temperature tetragonal
(rutile) phase via the intermediate phase was observed around 340 K, as
reported previously Kim . Considering the possibility that the sample already
contained hydrogen, the hydrogen content was observed by thermal desorption
spectrometry (TDS1200, ESCO) (Fig. S4). The amount of desorbed hydrogen was
1.14$\times$1019 cm-3 (corresponding to H0.00034VO2) upon increasing the
temperature to 1073 K. The magnetic susceptibility was measured using the
SQUID magnetometer (MPMS, Quantum Design, Inc.) from 5 to 400 K. No magnetic
anomalies were observed in this temperature range, except for the M-I
transition of VO2 [Fig. 2(a)]. Vanadium oxides have many analogs with slightly
different oxygen ratios that exhibit different magnetic transition
temperatures CVD . Based on these results, our sample quality is adequate for
studying the nature of VO2. $\mu$SR experiments were performed using the
ARTEMIS spectrometer installed in the S1 area at Muon Science Establishment
(MUSE), Japan Proton Accelerator Research Complex (J-PARC).

##  III Results and discussion

Figure 1(a) shows zero-field $\mu$SR time spectra at several temperatures from
5 K to 380 K. These spectra above $\sim$200 K show the Gaussian-like line
shapes mainly attributed to the random local fields from the nuclear magnetic
moment of 51V (5.148 $\mu_{\rm N}$). However, below 150 K, it can be seen that
a fast exponential relaxation with partial loss of asymmetry appears in the
initial time region ($t\leq\sim$1 $\mu$s). This suggests the gradual
development of microscopic magnetic order of vanadium electron spins in parts
of the sample, which may result from lattice imperfections LT29 .

Figure 1: Zero-field $\mu$SR time spectra in VO2 measured at various
temperatures (5 K$\sim$380 K). The solid lines represent the result of least-
squares fitting by Eq.1. All spectra were measured in the temperature
increasing process.

Based on the above observation, we analyzed the time spectra using the
following curve-fit function:

$G_{\mathrm{Z}}(t)=AG_{\mathrm{KT}}(\Delta,\nu,B_{\mathrm{ext}},t)e^{-\lambda
t}+A_{\mathrm{bg}}.\ \ (1)$

A is a signal amplitude called asymmetry, which reflects the volume fraction
of the sample. The first term on the right-hand side is the product of the
dynamic Gaussian Kubo-Toyabe function $G_{\rm{KT}}(\Delta,\nu,B_{\rm{ext}},t)$
Haya  which represents the relaxation due to random local fields exerted from
nuclear magnetic moments (described by nuclear dipole-field width $\Delta$,
hopping rate of the muon $\nu$, longitudinal field $B_{\rm{ext}}$), and the
exponential relaxation component due to hyperfine interaction with electron
spins (relaxation rate $\lambda$). The second term on the right-hand side
($A_{\rm{bg}}$) represents the $T$-independent background term from the
surrounding sample holder and cryostat walls. When $\Delta\geq\nu$, the muon
spin depolarization can be quenched by applying $B_{\rm ext}\approx$
2–3$\times\Delta/\gamma_{\mu}$, so a global fit of the spectra at different
$B_{\rm{ext}}$ (0, 5, 10 G) using Eq. (1) can determine $\Delta$ and $\nu$
reliably at each temperature (Fig. S5). The longitudinal field dependence of
the time spectrum is clearly observed even after 10 $\mu$s. This fact suggests
that the high-flux pulsed muon at J-PARC has enabled one to separate $\nu$ and
$\Delta$ in VO2 clearly for the first time. black

Figure 2: Temperature dependence of (a) the magnetic susceptibility $\chi$,
(b) the asymmetry $A$, (c) the relaxation rate $\lambda$, (d) the nuclear
dipole-field width $\Delta$, and (e) the hopping rate $\nu$. The inset in (e)
shows the Arrhenius plot of $\nu$ as a function of 1/$T$ with a line of best
fit from 200 K to 280 K. The temperature dependence of $\nu$ is well explained
by a thermal activation process, $\nu=\nu_{0}$exp(-$E_{a}/kT$), where
$\nu_{0}[$=24(3) $\mu$s-1] is the prefactor (pre-exponential) dependent on the
dipole-dipole coupling between muon and nuclear spins, and $E_{a}$[=0.11 (1)
eV] is the activation barrier for muon diffusion.

Figures 2 shows the temperature dependencies of the magnetic susceptibility
$\chi$ and the $\mu$SR fitting parameters $A$, $\lambda$, $\Delta$ and $\nu$.
$\chi$ exhibits a paramagnetic-nonmagnetic transition at $T_{\rm MI}$ with a
thermal hysteresis of 10 K, while there is no indication of any magnetic
anomalies below $T_{\rm MI}$ [Fig. 2(a)]. While this is seemingly inconsistent
with the decrease of $A$ (reflecting the volume fraction of the nonmagnetic
component) and associated increase in $\lambda$ below $\sim$150 K (indicating
the emergence of the local field from unpaired electron spins) [Fig. 2(b) and
(c)], our previous study indicates that this is due to the coexistence of the
singlet state and microscopic magnetic ordered states caused by structural
imperfections, such as lattice defects LT29 . The increase in $\lambda$ above
180 K is due to two possible reasons: (1) unpaired electrons are located
adjacent to the defect where muons are trapped, forming a polaron-like state
Shimop ; Vilaop ; Itop ; Dehnp1 ; Dehnp2 , and (2) $\lambda$ and nuclear
relaxation are not completely separated ($\lambda$ is affected by changes in
$\nu$) in the high-temperature region where $\nu\gg\Delta$. (A discussion of
magnetism is beyond the scope of this paper and will be omitted.)

$\Delta$ shows a gradual decrease with increasing temperature, markedly above
200 K [Fig. 2(d)]. Note that the spectra could not be reproduced by Eq. (1)
with $\Delta$ fixed to a constant (Fig. S6). This is not expected unless the
51V distribution varies due to changes in crystal structure or muon sites [see
Eq. (2) below]. Considering that the fluctuation rate $\nu$ increases in the
relevant temperature range [Fig. 2(e)], the decrease in $\Delta$ can be
attributed to the change of the muon site by diffusion; it is unlikely that
the diffusion of V ions ($\sim$500 times heavier than muons) sets in at such
low temperatures. The activation energy $E_{\rm a}$ of the hopping motion
estimated from the slope of the line in the Arrhenius plot for $\nu$ the inset
of Figs. 2(e), is approximately 0.11 eV in the temperature range from 200 K to
280 K. This value is comparable to the activation energy for hydrogen
diffusion (0.08-0.12 eV) in Sr/BaTiO(3-x)Hx hydride conductors Ito ; BTO . The
increase in $\nu$ begins to be observed around 180 K and reaches about 0.15
$\mu$s-1 around 250 K, which is similar to that reported for other oxides as a
result of $\mu^{+}$ diffusion Nishida ; Sonier . On the other hand, $\nu$ is
suppressed in the high-temperature side above $\sim$250 K and saturated at 280
K. This also correlates with a decrease in $\Delta$, suggesting that the
diffusion pathway may change near the structural phase transition. Thus, the
pre-exponential coefficients obtained by fitting the data, including 280 K
with the Arrhenius formula, are about an order of magnitude smaller than for
other oxides, which also supports such a change in the diffusion mechanism.
black

Figure 3: Calculated distribution of $\Delta$ for the cross-sectional layers
parallel or perpendicular to the oxygen channels in the tetragonal and the
monoclinic VO2. The values of $\Delta$ (numerical values in the contour map,
unit: Gauss) are calculated by using Dipelec code Koji  with reported lattice
parameters Eyert . Crystal structures were visualized using the VESTA program
Izumi .

Above room temperature, the increase in $\nu$ shows a sharp drop around
$T_{\rm MI}$ [Fig. 2(e) and Fig. S7(e)]. This behavior may attribute to the
trapping effect by lattice defects and phase boundaries. The muons, which
diffuse rapidly due to the temperature rise, are easily trapped in the lattice
defects that require considerable activation energy to escape Vilao .
Furthermore, since three phases coexist during the M-I transition of VO2, the
distribution of $E_{\rm a}$ is expected to be larger at the phase boundary due
to structural disorder associated with tweedy or glassy textures of mixed
phases with characteristic correlation lengths of $\sim$5 nm observed in VO2
epitaxial films twd . In addition, the carrier mobility measurement has
inferred a strong scattering of electrons at the boundaries of metallic and
insulating domains in the inhomogeneous transition of VO2 Fu . Given the above
facts, the strong lattice inhomogeneity in the coexisting state inherent to
the first-order phase transition contributes to the sharp drop of $\nu$ around
$T_{\rm MI}$. This is also supported by the fact that the dropping temperature
of $\nu$ roughly coincides with the appearance of the coexisting state (Fig.
S7).

To discuss the crystallographic sites of muon (or hydrogen) in VO2, we
performed the theoretical calculation of $\Delta$ versus muon position. Since
muon is an ultra-high sensitivity probe of local magnetic fields, we can
estimate the muon sites using a spatial configuration of 51V nuclear magnetic
moments as a guide. In polycrystals under zero magnetic field, $\Delta^{2}$
can be evaluated as the second moment of internal field distribution exerted
from nuclear magnetic moments ($I$ = 7/2 in 51V, which is subject to
quadrupole interaction) using the following formula Hemp :

$\Delta^{2}\approx\frac{4}{9}\gamma_{\mu}^{2}\left(\frac{\mu_{0}}{4\pi}\right)^{2}\sum\frac{\mu_{i}^{2}}{r_{i}^{6}},\
\ (2)$

where $\gamma_{\mu}$ (= 2$\pi\times$13.553 kHz/G) is the muon gyromagnetic
ratio, $r_{i}$ is the distance from the $i$-th nuclear magnetic moment
$\mu_{i}$, and $\mu_{0}$ is the vacuum permeability. Figure 3 shows the
calculated spatial distribution of $\Delta$ for the cross-sectional layers
parallel or perpendicular to the oxygen channels in the tetragonal and the
monoclinic structures. Comparison of the calculated results for both
structures shows that there is no significant difference in the $\Delta$
distribution within each oxygen channel. The value of $\Delta$/$\gamma_{\mu}$
is just under 4 G ($\Delta$$\sim$0.34 $\mu$s-1) at the center of the oxygen
channel and has similar values along the channel direction. The maximum value
of $\Delta$/$\gamma_{\mu}$ determined from experiments is about 3.2 G
($\Delta$$\sim$0.27 $\mu$s-1) at 4 K, indicating that the muons are located
near the center of the oxygen channel at low temperatures. This result is in
interesting agreement with the muon/hydrogen case in $\beta$-MnO2 which has a
similar structure to VO2 and is presumed to be a common property of rutile-
type oxides OKB .

As the temperature increases, $\Delta$ decreases rapidly to about one-fifth of
its low-temperature counterpart ($\Delta$$\sim$0.05 $\mu$s-1) above $T_{\rm
MI}$. The muon sites corresponding to such a small value of $\Delta$ are not
found for the interstitial position, even considering a lattice distortion
caused by hydrogen intercalation in the rutile structure. Therefore, it is
highly likely that implanted muons are trapped in vacancies at high
temperatures. Since the crystallite size of this sample is 50$\sim$80 nm (Fig.
S3), the surface contribution is expected to be small. However, muons might
have diffused to the grain boundaries at high temperatures considering the
saturation of $\nu$ above 280 K. black

We performed a simulation of the relationship between $\Delta$ and the size of
the vacancy based on Eq. (2). The virtual element muon (Mu) was fixed at the
center of the oxygen channel, and $\Delta$ was calculated for a variety of
situations where the V ions at the first, second, and third nearest neighbors
(V1,V2, and V3) were removed sequentially. Note that structural distortions
due to the vacancy was not taken into account. Figure 4(a) shows the reduction
rate of $\Delta$ as a function of the number of removed vanadium $N$. One can
see that $\Delta$ decreases almost linearly as $N$ increases within the each V
shell (V1-V3), down to about 20% when the V ions in the V3 shell (the third
nearest neighbors) are removed. Figure 4(b) compares the simulation result and
the experimental $\Delta$. The existence of the vacancy of a radius of
$\sim$0.38 nm (the distance between center Mu and V3) can explain the
magnitude of $\Delta$ above $T_{\rm MI}$. The discrepancy in $\Delta$ between
our result and that reported in the previous $\mu$SR study Mengyan  can be
interpreted as due to the difference in the oxygen ratios between samples.
According to their $\mu$SR result, some magnetic order was observed even in
the nonmagnetic state below $T_{\rm MI}$. While the details of the sample
quality in Ref. Mengyan  are unknown, magnetic secondary phases may have
formed during the bulk sintering process. In any case, our results strongly
suggest that muons at high temperatures spend most of their life in a
diffusive process of repeated ”capture and release” by vacancies. The $\mu$SR
study in a similar trapping mechanism has been discussed in semiconductors
Hemp ; Vilao . These two different muons (interstitial and vacancy) also exist
in $\beta$-MnO2 and correspond to two other hydrogens called Ruetschi and
Coleman protons in battery materials Ruet1 ; Ruet2 ; Colm .

Figure 4: (a) The reduction rate of $\Delta$ as a function of the number of
removed vanadium $N$. The inset shows the spatial arrangement of muon (Mu) and
surrounding vanadium, the first (V1), second (V2), and third (V3) nearest
neighbors. (b) Temperature dependence of the experimental $\Delta$ with the
simulation result. The dotted lines indicate the reduction rate of $\Delta$ at
each stage of removing V1, V2, and V3.

Finally, to gain further insight into the interstitial hydrogen dynamics in
VO2, we discuss muon diffusion in the oxygen channel at intermediate
temperatures. The diffusion coefficient of muon $D_{\mu}$ is approximated by
the following equation Borg :

$D_{\mu}=\sum_{i=1}^{n}\frac{1}{N_{i}}Z_{i}s_{i}^{2}\nu,\ \ (3)$

where $N_{i}$ is the number of muon hopping sites in the $i$-th path, $Z_{i}$
is the vacancy fraction, and $s_{i}$ is the hopping distance. The interstitial
hydrogen diffusion pathway is one-dimensional-like along the O-channel in VO2,
and the diffusion transverse to the channel can be negligible John ; Lin .
Assuming that muon diffusion via the O-Mu bond is dominant below room
temperature, a muon jumps to one of the four neighboring oxygen atoms
($N_{i}$=4, the average O-O distance $s_{i}$=0.271 nm). Additionally,
neglecting the existence of other muons or hydrogen ($Z_{i}$=1) and taking the
experimental value of the hopping rate at 280 K [$\nu$=0.20(2) $\mu$s-1] to
Eq. (3) yields $D_{\mu}$=1.47(2)$\times$10-10 cm2/s. This value is on the same
order of magnitude as the reported diffusion constants of hydrogen
$D_{\mu}$=0.46$\times$10-10 cm2/s at 300 K Mura  and
$D_{\mu}$=6.7$\times$10-10 cm2/s at 373 K Lin , indicating that the result of
our hydrogen simulation by muon is reliable; the mass difference (the muon’s
mass is 1/9th that of the proton) is unimportant at high temperatures where
only moderate isotope dependences, e.g., $D_{\mu}$=4.7(9)$\times$10-4 cm2/s
and $D_{\mu}$=7.5$\times$10-4 cm2/s above 300 K in iron Gpan , are expected.
Interestingly, it is about three orders of magnitude larger than $D_{\mu}$
(=1.8$\times$10-13 cm2/s at 298 K) in TiO2, which has the same crystal
structure John . Based on the above results, we conclude that the interstitial
hydrogen in VO2 has the potential to diffuse at a sufficient rate (i.e.,
$\sim$10-10 cm2/s) even at room temperature, suggesting that it is suitable
for applications requiring a fast response, such as ReRAM. However, high-
quality VO2 thin films are required to fabricate high-reliability devices
because defect/vacancy-mediated muon diffusion is observed even below room
temperature [Figure 4(b)].

##  IV Summary

In summary, we have provided nano-scale information on the hydrogen dynamics
in VO2 using muon as pseudo-hydrogen. The temperature dependence of the muon
hopping rate and the spatial distribution of the 51V nuclear magnetic moments
indicates the existence of two types of muon diffusion (via O-Mu bonds and
defects/vacancies) and that a high-quality thin film can have a diffusion
coefficient of the order of 10-10 cm2/s around room temperature. Our findings
will provide a useful criterion for designing hydrogen-driven electronic
devices in the future.

## Acknowledgment

This work was supported by the MEXT Elements Strategy Initiative to Form Core
Research Centers, from the Ministry of Education, Culture, Sports, Science,
and Technology of Japan (MEXT) under Grant No. JPMXP0112101001 and partially
supported by the MEXT Program: Data Creation and Utilization Type Material
Research and Development Project under Grant No. JPMXP1122683430 and JSPS
KAKENHI (Grant No. 20K05312, 19H05819). The $\mu$SR experiments were performed
under user programs (Proposal No. 2019MS02) at the Materials and Life Science
Experimental Facility of the J-PARC. We also acknowledge the Neutron Science
and Technology Center, CROSS for the use of MPMS in their user laboratories.

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# Abstract embeddability ranks

Florent P. Baudier  Department of Mathematics
Texas A&M University
College Station
Texas
USA<EMAIL_ADDRESS>https://www.math.tamu.edu/ florent/ and  Christian
Rosendal  Department of Mathematics
University of Maryland
4176 Campus Drive - William E. Kirwan Hall
College Park, MD 20742-4015
USA<EMAIL_ADDRESS>sites.google.com/view/christian-rosendal/

###### Abstract.

We describe several ordinal indices that are capable of detecting, according
to various metric notions of faithfulness, the embeddability between pairs of
Polish spaces. These embeddability ranks are of theoretical interest but seem
difficult to estimate in practice. Embeddability ranks, which are easier to
estimate in practice, are embeddability ranks generated by Schauder bases.
These embeddability are inspired by the nonlinear indices à la Bourgain from
[BLMS_FM]. In particular, we resolve a problem [BLMS_FM, Problem 3.10]
regarding the necessity of additional set-theoretic axioms regarding the main
coarse universality result of [BLMS_FM].

The authors were partially supported by the U.S. National Science Foundation
under Grant Numbers DMS-2055604 (F.B.) and DMS-2204849 (C.R.).

##  1\. Introduction

In 1980, Bourgain [Bourgain1980] showed that a separable Banach space that
contains an isomorphic copy of every separable _reflexive_ space must also
contain an isomorphic copy of $C[0,1]$, and thus, by Banach-Mazur embedding
theorem, an isomorphic copy of every separable Banach space. This universality
theorem is a far reaching improvement of an earlier result of Szlenk
[Szlenk1968], which states that there is no separable reflexive Banach space
that is isomorphically universal for the class of separable reflexive Banach
spaces. Bourgain’s argument in turn relies on a reformulation of the problem
in terms of the computation of an ordinal index, making the problem amenable
to descriptive set theoretic techniques. Bourgain’s innovative approach was
further developed in [Bossard2002], [ArgyrosDodos], and [Dodos2009], in order
to show that a class of Banach spaces that is analytic, in the Effros-Borel
structure of subspaces of $C[0,1]$, and contains all separable reflexive
Banach spaces, must contain a universal space. In [BLMS_FM], nonlinear
versions of Bourgain’s universality theorem were proposed. Thanks to Aharoni’s
theorem [Aharoni1974], it is known that $c_{0}$ contains a bi-Lipschitz, and
hence a coarse and uniform, copy of every separable metric space. Depending on
whether we consider bi-Lipschitz embeddings or coarse embeddings, the role of
the class of reflexive spaces considered by Szlenk or Bourgain is played by a
collection of metric spaces $R[\mathcal{S}_{\alpha}]_{c_{0}}$ with
${\alpha<\omega_{1}}$ and where $R=\mathbb{Q}$ for bi-Lipschitz embeddings and
$R=\mathbb{Z}$ for coarse embeddings. Here $R[\mathcal{S}_{\alpha}]_{c_{0}}$
is the metric subspace of $c_{0}$ consisting of all vectors $\sum_{i\in
A}x_{i}e_{i}$ where $x_{i}\in R$ and $A$ belong to the $\alpha^{\rm{th}}$
Schreier family $\mathcal{S}_{\alpha}$.

In this note we prove the coarse universality result below. This result was
originally proved in [BLMS_FM] under additional set theoretical assumptions,
namely Martin’s Axiom (MA) and the negation of the Continuum Hypothesis
($\neg\text{CH}$). The main point of our theorem is to show that these axioms
are actually not required, thus solving Problem 3.10 from [BLMS_FM].

######  Theorem.

If a separable metric space contains a coarse copy of
$\mathbb{Z}[\mathcal{S}_{\alpha}]_{c_{0}}$ for every ordinal
$\alpha<\omega_{1}$, then it contains a coarse copy of every separable metric
space.

In [BLMS_FM], the proof (under MA+$\neg$CH) of the theorem above relied on a
reformulation à la Bourgain of the universality problem. In particular,
certain combinatorial trees on Polish spaces capturing the nonlinearity of the
problem together with associated ordinal indices were introduced. In order for
this approach to be successful in the coarse setting, it was necessary to
extract an uncountable family of _equi_ -coarse embeddings out of an
uncountable family of merely coarse embeddings. While the similar problem in
the Lipschitz or linear settings can be solved with a simple pigeonhole
principle, the argument for the coarse case required the additional set
theoretic assumptions mentioned above.

######  Proposition 1.1 (MA+$\neg$CH).

If $(X_{\alpha},d_{\alpha})_{\alpha<\omega_{1}}$ is a collection of metric
spaces such that for all $\alpha<\omega_{1}$, $X_{\alpha}$ embeds coarsely
into a metric space $(M,d)$, then there exists an uncountable subset $C$ of
$\omega_{1}$ such that $(X_{\alpha},d_{\alpha})_{\alpha\in C}$ embeds equi-
coarsely into $(M,d)$.

In Section 2, we recall the definitions of the objects being considered in
this article (e.g. coarse, uniform, bi-Lipschitz embeddings) and introduce
various ordinal indices related to the corresponding notions of metric
embeddability. Our point of view relies on the notion of rank of well-founded
binary relations on Polish spaces (instead of trees on Polish spaces as in
[BLMS_FM]). In order to avoid the use of additional axioms through Proposition
1.1, we incorporate the main characteristics of the metric embeddings into the
definitions of the ordinal indices. The guaranteed uniform control on the
moduli in the conclusion of Proposition 1.1 becomes an automatic feature of
the definition and consequently the use of additional set theoretic axioms is
bypassed. In Section 2.2, we introduce some natural and easily defined
embeddability ranks. However, estimating these ranks does not seem
straightforward in practice and therefore, in Section 2.3, we define
embeddability ranks induced by Schauder bases. These ranks are similar to the
nonlinear indices introduced in [BLMS_FM] but are more general and most
importantly, as hinted at above, incorporate the compression and expansion
moduli into their definitions. As in [BLMS_FM], these ranks are easily
estimated and lead to the same results without the use of additional set
theoretic assumptions.

The extraction of subcollections of embeddings with uniform control on their
moduli of expansion and compression is an interesting problem in its own right
and, in the last section, we generalize Proposition 1.1 by relating this type
of problems to certain cardinal invariants of the continuum, in particular the
bounding number $\mathfrak{b}$. Our proof is arguably more transparent than
the one in [BLMS_FM].

##  2\. Embeddability ranks

###  2.1. Functional moduli

The various notions of embeddability we will consider can all be described in
terms of functional moduli, which we now proceed to define. For this purpose,
let

$\mathcal{Z}=\\{\ldots,\tfrac{1}{3},\tfrac{1}{2},1,2,3,\ldots\\}$

and set

$\mathcal{N}=\\{\kappa\colon\mathcal{Z}\to[0,\infty]\;\big{|}\;\kappa\text{ is
non-decreasing }\\}.$

Observe that $\mathcal{N}$ can be seen as a closed subset of the Polish space
$[0,\infty]^{\mathcal{Z}}$, where we view $[0,\infty]$ as the one-point
compactification of $[0,\infty)$.

When $\phi\colon X\to E$ is a function between two non-empty metric spaces, we
define the compression modulus $\kappa_{\phi}\in\mathcal{N}$ and the expansion
modulus $\omega_{\phi}\in\mathcal{N}$ by

$\kappa_{\phi}(r)=\inf\\{d_{E}\big{(}\phi(x),\phi(y)\big{)}\;\big{|}\;d_{X}(x,y)\geqslant
r\\},\quad r\in\mathcal{Z}$

and

$\omega_{\phi}(r)=\sup\\{d_{E}\big{(}\phi(x),\phi(y)\big{)}\;\big{|}\;d_{X}(x,y)\leqslant
r\\},\quad r\in\mathcal{Z}.$

Remark that, if $X$ is bounded, there may be no $x,y\in X$ for which
$d_{X}(x,y)\geqslant r$, which means that
$\kappa_{\phi}(r)=\inf\emptyset=\infty$. So, for any $r\in\mathcal{Z}$,

$\kappa_{\phi}(r)<\infty\;\Leftrightarrow\;\exists x,y\in
X\;\;d_{X}(x,y)\geqslant r.$

We need one more piece of notation. For any real number $t>0$, let $t_{-}$ and
$t_{+}$ be respectively the largest and smallest element of $\mathcal{Z}$ so
that

$t_{-}\leqslant t\leqslant t_{+}$

and observe that, for all $t>0$, we have $\frac{t}{2}\leqslant t_{-}\leqslant
t\leqslant t_{+}\leqslant 2t$. By definition of $\kappa_{\phi}$ and
$\omega_{\phi}$, we note that

$\kappa_{\phi}\big{(}d(x,y)_{-}\big{)}\leqslant
d\big{(}\phi(x),\phi(y)\big{)}\leqslant\omega_{\phi}\big{(}d(x,y)_{+}\big{)}$

for all $x,y\in X$.

As noted above, the various notions of embeddability between metric spaces we
shall consider are defined by imposing different restrictions of the
compression and expansion moduli of a function $\phi\colon X\to E$. We
therefore define

$\mathcal{M}_{\text{coarse}}=\\{(\kappa,\omega)\in\mathcal{N}\times\mathcal{N}\;\big{|}\;\omega(r)<\infty\text{
for all }r\in\mathcal{Z}\text{ and }\lim_{n\to\infty}\kappa(n)=\infty\\},$
$\mathcal{M}_{\text{uniform}}=\\{(\kappa,\omega)\in\mathcal{N}\times\mathcal{N}\;\big{|}\;0<\kappa(r)\text{
for all }r\in\mathcal{Z}\text{ and
}\lim_{n\to\infty}\omega(\tfrac{1}{n})=0\\}$

and finally

$\mathcal{M}_{\text{Lipschitz}}=\\{(\kappa,\omega)\in\mathcal{N}\times\mathcal{N}\;\big{|}\;\kappa(r)\geqslant
cr,\;\omega(r)\leqslant Cr\text{ for some }0<c<C<\infty\\}.$

A map $\phi\colon X\to E$ is then called a coarse, uniform or bi-Lipschitz
embedding according to whether $(\kappa_{\phi},\omega_{\phi})$ belongs to
$\mathcal{M}_{\text{coarse}}$, $\mathcal{M}_{\text{uniform}}$ or to
$\mathcal{M}_{\text{Lipschitz}}$.

###  2.2. Simple embeddability ranks

Suppose $X$ and $E$ are metric spaces and that $D=\\{x_{1},x_{2},\ldots\\}$ is
a fixed enumeration of a countable dense subset of $X$. We then let
$\Omega\big{(}{D,E}\big{)}$ be the collection of all quadruples
$(\kappa,\omega,k,\phi)\in\mathcal{N}\times\mathcal{N}\times\mathbb{N}\times
E^{D}$ so that

$\kappa\big{(}d(x,y)_{-}\big{)}\leqslant
d\big{(}\phi(x),\phi(y)\big{)}\leqslant\omega\big{(}d(x,y)_{+}\big{)}$

for all distinct $x,y\in\\{x_{1},x_{2},\ldots,x_{k}\\}$. Let also

${\mathcal{C}oa}\big{(}{D,E}\big{)}=\\{(\kappa,\omega,k,\phi)\in\Omega\big{(}{D,E}\big{)}\;\big{|}\;(\kappa,\omega)\in\mathcal{M}_{\text{coarse}}\\},$
${\mathcal{U}ni}\big{(}{D,E}\big{)}=\\{(\kappa,\omega,k,\phi)\in\Omega\big{(}{D,E}\big{)}\;\big{|}\;(\kappa,\omega)\in\mathcal{M}_{\text{uniform}}\\},$
${\mathcal{L}ip}\big{(}{D,E}\big{)}=\\{(\kappa,\omega,k,\phi)\in\Omega\big{(}{D,E}\big{)}\;\big{|}\;(\kappa,\omega)\in\mathcal{M}_{\text{Lipschitz}}\\}.$

Finally, we define a binary relation $<$ on $\Omega\big{(}{D,E}\big{)}$ by
letting

$\begin{split}{(\kappa,\omega,k,\phi)<&(\kappa^{\prime},\omega^{\prime},k^{\prime},\phi^{\prime})\;\Leftrightarrow\;\\\
&(\kappa,\omega)=(\kappa^{\prime},\omega^{\prime})\;\;\&\;\;k=k^{\prime}+1\;\;\&\;\;\phi\upharpoonright_{\\{x_{1},x_{2},\ldots,x_{k^{\prime}}\\}}=\phi^{\prime}\upharpoonright_{\\{x_{1},x_{2},\ldots,x_{k^{\prime}}\\}},}\end{split}$

where $\psi\upharpoonright_{A}$ denotes the restriction of a map $\psi$ to a
subset $A$.

Recall that a binary relation $\prec$ on a set $A$ is said to be ill-founded
if and only if there is an infinite sequence $a_{1},a_{2},\ldots\in A$ so that

$\cdots\prec a_{3}\prec a_{2}\prec a_{1}.$

On the other hand, if there is no such infinite sequence, we say that $\prec$
is well-founded. Observe that another way of stating that $\prec$ is well-
founded is by saying that every non-empty subset $B\subseteq A$ has a minimal
element $x\in B$, i.e., so that $y\not\prec x$ for all $y\in B$.

######  Lemma 2.1.

The binary relation $<$ is ill-founded on any of the spaces

${\mathcal{C}oa}\big{(}{D,E}\big{)},\qquad{\mathcal{U}ni}\big{(}{D,E}\big{)},\qquad{\mathcal{L}ip}\big{(}{D,E}\big{)}$

if and only if $X$ coarsely embeds, uniformly embeds, respectively bi-
Lipschitz embeds into $E$.

###### Proof.

We give the proof for the case of coarse embeddings, the two other cases being
entirely analogous. So suppose first that $\phi\colon X\to E$ is a coarse
embedding and hence that
$(\kappa_{\phi},\omega_{\phi})\in\mathcal{M}_{\text{coarse}}$. Furthermore,

$\kappa_{\phi}\big{(}d(x,y)_{-}\big{)}\leqslant
d\big{(}\phi(x),\phi(y)\big{)}\leqslant\omega_{\phi}\big{(}d(x,y)_{+}\big{)}$

for all distinct $x,y\in X$. Letting $\psi=\phi\upharpoonright_{D}$, we have
$(\kappa_{\phi},\omega_{\phi},k,\psi)\in{\mathcal{C}oa}\big{(}{D,E}\big{)}$
for all $k$ and

$\dots<(\kappa_{\phi},\omega_{\phi},3,\psi)<(\kappa_{\phi},\omega_{\phi},2,\psi)<(\kappa_{\phi},\omega_{\phi},1,\psi)$

is an infinite descending sequence in ${\mathcal{C}oa}\big{(}{D,E}\big{)}$,
showing ill-foundedness.

Conversely, suppose that $<$ is ill-founded on the set
${\mathcal{C}oa}\big{(}{D,E}\big{)}$. This means that there is some
$(\kappa,\omega)\in\mathcal{M}_{\text{coarse}}$, a natural number $k_{0}$, and
functions $\phi_{k}\colon D\to E$ for $k\geqslant k_{0}$ so that, for all
$k\geqslant k_{0}$,

$\phi_{k+1}\upharpoonright_{\\{x_{1},x_{2},\ldots,x_{k}\\}}=\phi_{k}\upharpoonright_{\\{x_{1},x_{2},\ldots,x_{k}\\}}$

and

$\kappa\big{(}d_{X}(x,y)_{-}\big{)}\leqslant
d_{E}\big{(}\phi_{k}(x),\phi_{k}(y)\big{)}\leqslant\omega\big{(}d_{X}(x,y)_{+}\big{)}$

for all distinct $x,y\in\\{x_{1},x_{2},\ldots,x_{k}\\}$. It follows that there
is a single function $\phi\colon D\to E$ extending the restrictions
$\phi_{k}\upharpoonright_{\\{x_{1},x_{2},\ldots,x_{k}\\}}$ and satisfying

$\kappa\big{(}d_{X}(x,y)_{-}\big{)}\leqslant
d_{E}\big{(}\phi(x),\phi(y)\big{)}\leqslant\omega\big{(}d_{X}(x,y)_{+}\big{)}$

for all distinct $x,y\in D$. In particular, for all $n\geqslant 1$ and $x,y\in
D$, we have that

$d_{X}(x,y)\geqslant
n\;\Rightarrow\;d_{E}\big{(}\phi(x),\phi(y)\big{)}\geqslant\kappa\big{(}d_{X}(x,y)_{-}\big{)}\geqslant\kappa(n)$

and

$d_{X}(x,y)\leqslant
n\;\Rightarrow\;d_{E}\big{(}\phi(x),\phi(y)\big{)}\leqslant\omega\big{(}d_{X}(x,y)_{+}\big{)}\leqslant\omega(n).$

Because $D$ is dense in $X$, we can now extend $\phi$ to a map $\phi\colon
X\to E$ as follows. For $x\in X\setminus D$, pick an arbitrary point
$x^{\prime}\in D$ so that $d_{X}(x^{\prime},x)<\frac{1}{2}$ and set

$\phi(x)=\phi(x^{\prime}).$

Then we still have

$d_{X}(x,y)\geqslant
n+1\;\Rightarrow\;d_{E}\big{(}\phi(x),\phi(y)\big{)}\geqslant\kappa(n)$

and

$d_{X}(x,y)\leqslant
n-1\;\Rightarrow\;d_{E}\big{(}\phi(x),\phi(y)\big{)}\leqslant\omega(n).$

As $\lim_{n\to\infty}\kappa(n)=\infty$ and $\omega(n)<\infty$ for all $n$, it
follows that $\phi$ is a coarse embedding. ∎

Recall that, if $\prec$ is a well-founded relation on a set $A$, we may define
an ordinal-valued rank function $\rho\colon A\to{\sf Ord}$ by letting

$\rho(x)=\sup\\{\rho(y)+1\;\big{|}\;y\prec x\\}$

for all $x\in A$. Thus, $\rho(x)=0$ if and only if $x\in A$ is minimal, that
is, if $y\not\prec x$ for all $y\in A$. Similarly, $\rho(x)\geqslant\omega$ if
and only if, for all $m\geqslant 1$, one may find $x_{1},\ldots,x_{m}\in A$ so
that

$x_{1}\prec x_{2}\prec\dots\prec x_{m}\prec x.$

The image $\rho[A]=\\{\rho(x)\;\big{|}\;x\in A\\}$ of the rank function $\rho$
associated with a particular well-founded relation $\prec$ on a set $A$ is
always an initial segment of the ordinals and hence is an ordinal itself. That
allows us to define the rank by

${\sf rk}(A,\prec)=\rho[A]=\sup\\{\rho(x)+1\;\big{|}\;x\in A\\}\in{\sf Ord}.$

Let us now recall the well-known boundedness theorem for analytic well-founded
relations (see [Kechris, Theorem 31.1]). For this, let us say that a binary
relation $\prec$ on a Polish space $A$ is analytic if it is analytic when
viewed as a subset of $A\times A$.

######  Theorem 2.2.

Suppose $\prec$ is an analytic, well-founded, binary relation on a Borel
subset of a Polish space $A$. Then the rank ${\sf rk}(A,\prec)$ is countable.

The following theorem then connects Theorem 2.2 with Lemma 2.1.

######  Theorem 2.3.

Suppose $X$ and $E$ are separable complete metric spaces and
$D=\\{x_{1},x_{2},\ldots\\}$ is a fixed enumeration of a dense subset of $X$.
Then either $X$ coarsely embeds into $E$ or the rank

${\sf rk}\big{(}{\mathcal{C}oa}\big{(}{D,E}\big{)},<\big{)}$

is countable.

Similar results hold for uniform and bi-Lipschitz embeddability, and our proof
applies to all of the three cases.

###### Proof.

Because $D$ is countable and $E$ is Polish, the set $\Omega\big{(}D,E\big{)}$
is a closed subset of the Polish space
$\mathcal{N}\times\mathcal{N}\times\mathbb{N}\times E^{D}$. Also,
$\mathcal{M}_{\text{coarse}}$, $\mathcal{M}_{\text{uniform}}$, and
$\mathcal{M}_{\text{Lipschitz}}$, are all Borel sets in
$\mathcal{N}\times\mathcal{N}$, whereby each of the sets

${\mathcal{C}oa}\big{(}{D,E}\big{)},\qquad{\mathcal{U}ni}\big{(}{D,E}\big{)},\qquad{\mathcal{L}ip}\big{(}{D,E}\big{)}$

are Borel subsets of the Polish space
$\mathcal{N}\times\mathcal{N}\times\mathbb{N}\times E^{D}$. Furthermore, the
relation $<$ is closed as a subset of $\Omega\big{(}D,E\big{)}$. The theorem
now follows from Theorem 2.2 and Lemma 2.1. ∎

###  2.3. Embeddability ranks based on Schauder bases

In this section we define embeddability ranks that are a little more delicate
to define but are easier to estimate from below.

Let $R$ be a ring and denote by $R[e_{1},e_{2},\ldots]$ the free $R$-module
with basis elements $e_{1},e_{2},\ldots$. Also, if
$A\subseteq\\{e_{1},e_{2},\ldots\\}$, we let $R[A]$ denote the submodule of
$R[e_{1},e_{2},\ldots]$ spanned by $A$. For two finite subsets
$A,B\subseteq\\{e_{1},e_{2},\ldots\\}$, we write

$A\prec^{\star}B$

to denote that $B=\\{e_{n_{1}},\ldots,e_{n_{k}}\\}$ and
$A=\\{e_{n_{1}},\ldots,e_{n_{k}},e_{n_{k+1}}\\}$ for some
$n_{1}<\ldots<n_{k}<n_{k+1}$.

Suppose now that $(e_{n})_{n=1}^{\infty}$ is a basic sequence in a real Banach
space $(X,\lVert\cdot\rVert)$. Let also the ring $R$ be either $\mathbb{Z}$ or
$\mathbb{Q}$, in which case, $R[e_{1},e_{2},\ldots]\subseteq X$. We thus view
$R[e_{1},e_{2},\ldots]$ as a metric space with the metric induced by the norm
on $X$. Assume that $E$ is a metric space and define
$\Theta\big{(}R,(e_{n}),E)$ as the collection of all quadruples
$(\kappa,\omega,A,\phi)$ where $\kappa,\omega\in\mathcal{N}$, $A$ is a finite
subset of $\\{e_{1},e_{2},\ldots\\}$ and $\phi$ is a function
$R[e_{1},e_{2},\ldots]\mathop{\overset{\phi}{\longrightarrow}}E$ so that

$\kappa\big{(}\lVert x-y\rVert_{-}\big{)}\leqslant
d_{E}\big{(}\phi(x),\phi(y)\big{)}\leqslant\omega\big{(}\lVert
x-y\rVert_{+}\big{)}$

for all distinct $x,y\in R[A]$. As in the preceding section, we let

${\mathcal{C}oa}\big{(}{R,(e_{n}),E}\big{)}=\\{(\kappa,\omega,A,\phi)\in\Theta\big{(}R,(e_{n}),E\big{)}\;\big{|}\;(\kappa,\omega)\in\mathcal{M}_{\text{coarse}}\\},$
${\mathcal{U}ni}\big{(}{R,(e_{n}),E}\big{)}=\\{(\kappa,\omega,A,\phi)\in\Theta\big{(}R,(e_{n}),E\big{)}\;\big{|}\;(\kappa,\omega)\in\mathcal{M}_{\text{uniform}}\\},$
${\mathcal{L}ip}\big{(}{R,(e_{n}),E}\big{)}=\\{(\kappa,\omega,A,\phi)\in\Theta\big{(}R,(e_{n}),E\big{)}\;\big{|}\;(\kappa,\omega)\in\mathcal{M}_{\text{Lipschitz}}\\}$

and define a binary relation $<$ on $\Theta\big{(}R,(e_{n}),E\big{)}$ by
setting

$\begin{split}{(\kappa,\omega,A,\phi)<(\kappa^{\prime},\omega^{\prime},A^{\prime},&\phi^{\prime})\;\Leftrightarrow\;\\\
&(\kappa,\omega)=(\kappa^{\prime},\omega^{\prime})\;\;\&\;\;A\prec^{\star}A^{\prime}\;\;\&\;\;\phi\upharpoonright_{R[A^{\prime}]}=\phi^{\prime}\upharpoonright_{R[A^{\prime}]}.}\end{split}$

In analogy we Lemma 2.1, we now have the following.

######  Lemma 2.4.

The binary relation $<$ is ill-founded on any of the spaces

${\mathcal{C}oa}\big{(}{R,(e_{n}),E}\big{)},\qquad{\mathcal{U}ni}\big{(}{R,(e_{n}),E}\big{)},\qquad{\mathcal{L}ip}\big{(}{R,(e_{n}),E}\big{)}$

if and only if there is an infinite subsequence
$e_{n_{1}},e_{n_{2}},e_{n_{3}},\ldots$ of the basis of $X$ so that
$R[e_{n_{1}},e_{n_{2}},e_{n_{3}},\ldots]$ coarsely embeds, uniformly embeds,
respectively bi-Lipschitzly embeds into $E$.

Now, when $(e_{n})_{n=1}^{\infty}$ is a subsymmetric Schauder basis for $X$,
then, for any infinite subsequence $e_{n_{1}},e_{n_{2}},e_{n_{3}},\ldots$, the
map $e_{k}\mapsto e_{n_{k}}$ extends to a bi-Lipschitz equivalence between the
two metric spaces $R[e_{1},e_{2},e_{3},\ldots]$ and
$R[e_{n_{1}},e_{n_{2}},e_{n_{3}},\ldots]$. Also, as $R$ is countable, so is
$R[e_{1},e_{2},e_{3},\ldots]$. The following result is thus obtained exactly
as Theorem 2.3.

######  Theorem 2.5.

Suppose $(e_{n})_{n=1}^{\infty}$ is a subsymmetric Schauder basis for a Banach
space $X$, $E$ is a separable complete metric space and $R$ is either
$\mathbb{Z}$ or $\mathbb{Q}$. Then either $R[e_{1},e_{2},e_{3},\ldots]$
coarsely embeds into $E$ or the rank

${\sf rk}\big{(}{\mathcal{C}oa}\big{(}{R,(e_{n}),E}\big{)},<\big{)}$

is countable.

Again, similar results hold for uniform and bi-Lipschitz embeddability.

Observe now that, if $R=\mathbb{Q}$, then $R[e_{1},e_{2},e_{3},\ldots]$ is a
dense subspace of the Banach space $X$ and so coarse, uniform or bi-Lipschitz
embeddability of $X$ into $E$ is equivalent to the corresponding embeddability
of $R[e_{1},e_{2},e_{3},\ldots]$ into $E$. We thus have the following
corollary.

######  Corollary 2.6.

Suppose $(e_{n})_{n=1}^{\infty}$ is a subsymmetric Schauder basis for a Banach
space $X$ and $E$ is a separable complete metric space. Then either $X$
coarsely embeds into $E$ or the rank

${\sf rk}\big{(}{\mathcal{C}oa}\big{(}{\mathbb{Q},(e_{n}),E}\big{)},<\big{)}$

is countable.

In the specific case when $(e_{n})_{n=1}^{\infty}$ is the standard unit vector
basis for $X=c_{0}$, we note that $\mathbb{Z}[e_{1},e_{2},e_{3},\ldots]$ is
$1$-dense in $c_{0}$ and hence that coarse embeddability of $c_{0}$ into $E$
is again equivalent to that of $\mathbb{Z}[e_{1},e_{2},e_{3},\ldots]$ into
$E$.

######  Corollary 2.7.

Suppose $(e_{n})_{n=1}^{\infty}$ is the standard unit vector basis for $c_{0}$
and $E$ is a separable complete metric space. Then either $c_{0}$ coarsely
embeds into $E$ or the rank

${\sf rk}\big{(}{\mathcal{C}oa}\big{(}{\mathbb{Z},(e_{n}),E}\big{)},<\big{)}$

is countable.

###  2.4. Schreier spaces

To provide lower estimates for the coarse embeddability ranks, as in [BLMS_FM]
we will employ metric spaces built over the so-called Schreier sets, which are
compact hereditary families of finite subsets of
$\mathbb{N}=\\{1,2,3,\ldots\\}$. For this, if $A$ and $B$ are finite subsets
of $\mathbb{N}$ and $n\in\mathbb{N}$, we write

$n\leqslant A$

to denote that either $A$ is empty or that $n\leqslant\min A$ and write

$A<B$

to mean that either one of $A$ and $B$ is empty or that $\max A<\min B$.
Finally, let

$A\prec^{\star}B$

denote that $B=\\{{n_{1}},\ldots,{n_{k}}\\}$ and
$A=\\{{n_{1}},\ldots,{n_{k}},{n_{k+1}}\\}$ for some
$n_{1}<\ldots<n_{k}<n_{k+1}$. For every countable limit ordinal $\lambda$, we
fix once and for all an increasing sequence
$\lambda_{1}<\lambda_{2}<\lambda_{3}<\ldots$ of ordinals with limit $\lambda$.

By induction on $\alpha<\omega_{1}$, we define a family
$\mathcal{S}_{\alpha}\subseteq\mathcal{P}(\mathbb{N})$ consisting exclusively
of finite subsets of $\mathbb{N}$ so that $\mathcal{S}_{\alpha}$ is
hereditary, i.e., if $A\subseteq B$ and $B\in\mathcal{S}_{\alpha}$, then
$A\in\mathcal{S}_{\alpha}$, and so that $\mathcal{S}_{\alpha}$ is compact when
viewed as a subset of the Cantor space $\\{0,1\\}^{\mathbb{N}}$. This is done
as follows.

$\begin{split}{\mathcal{S}_{0}&=\big{\\{}\\{n\\}\;\big{|}\;n\in\mathbb{N}\big{\\}}\cup\\{\emptyset\\},\\\
\mathcal{S}_{\alpha+1}&=\big{\\{}\bigcup_{j=1}^{n}A_{j}\;\big{|}\;n\in\mathbb{N}\;\;\&\;\;A_{1},\ldots,A_{n}\in\mathcal{S}_{\alpha}\;\;\&\;\;n\leqslant
A_{1}<\ldots<A_{n}\big{\\}},\\\
\mathcal{S}_{\lambda}&=\big{\\{}A\;\big{|}\;\exists
n\in\mathbb{N}\;\;\;A\in\mathcal{S}_{\lambda_{n}}\;\&\;n\leqslant
A\big{\\}}\text{ for $\lambda$ a limit ordinal}.}\end{split}$

By transfinite induction on $\alpha<\omega_{1}$, one can show that

${\sf rk}\big{(}\mathcal{S}_{\alpha},\prec^{\star}\big{)}=\omega^{\alpha}+1.$

If now $(e_{n})_{n=1}^{\infty}$ is a Schauder basis for a Banach space $X$, we
let

$\mathbb{Z}[\mathcal{S}_{\alpha}]=\Big{\\{}\sum_{i\in
A}t_{i}e_{i}\;\Big{|}\;A\in\mathcal{S}_{\alpha}\;\;\&\;\;t_{i}\in\mathbb{Z}\Big{\\}}.$

Note that, contrary to the subspaces $\mathbb{Z}[A]$ for
$A\subseteq\\{e_{1},e_{2},\ldots\\}$, the subset
$\mathbb{Z}[\mathcal{S}_{\alpha}]$ will not be a submodule of
$\mathbb{Z}[e_{1},e_{2},\ldots]$ as it is not closed under addition. We write
$\mathbb{Z}[\mathcal{S}_{\alpha}]_{X}$ for the metric space obtained by
endowing $\mathbb{Z}[\mathcal{S}_{\alpha}]$ with the distance induced by the
norm of $X$.

######  Theorem 2.8.

Suppose $(e_{n})_{n=1}^{\infty}$ is the standard unit vector basis for $c_{0}$
and $E$ is a separable complete metric space. Then $c_{0}$ coarsely embeds
into $E$ if and only if $\mathbb{Z}[\mathcal{S}_{\alpha}]_{c_{0}}$ coarsely
embeds into $E$ for all $\alpha<\omega_{1}$.

###### Proof.

Suppose $\alpha<\omega_{1}$ and that
$\mathbb{Z}[\mathcal{S}_{\alpha}]_{c_{0}}\mathop{\overset{\phi}{\longrightarrow}}E$
is a coarse embedding and let $\kappa_{\phi}$ and $\omega_{\phi}$ be the
compression and expansion moduli of $\phi$. Extend also $\phi$ arbitrarily to
a map
$\mathbb{Z}[e_{1},e_{2},\ldots]\mathop{\overset{\phi}{\longrightarrow}}E$.
This means that, for all $A\in\mathcal{S}_{\alpha}$, we have

$\kappa_{\phi}\big{(}\lVert x-y\rVert_{-}\big{)}\leqslant
d_{E}\big{(}\phi(x),\phi(y)\big{)}\leqslant\omega_{\phi}\big{(}\lVert
x-y\rVert_{+}\big{)}$

whenever $x,y\in\mathbb{Z}[\\{e_{i}\\}_{i\in A}]$ are distinct and so

$\big{(}\kappa_{\phi},\omega_{\phi},\\{e_{i}\\}_{i\in
A},\phi\big{)}\in{\mathcal{C}oa}\big{(}{\mathbb{Z},(e_{n}),E}\big{)}.$

Furthermore, we note that the map
$\mathcal{S}_{\alpha}\mathop{\overset{f}{\longrightarrow}}{\mathcal{C}oa}\big{(}{\mathbb{Z},(e_{n}),E}\big{)}$
given by

$f(A)=\big{(}\kappa_{\phi},\omega_{\phi},\\{e_{i}\\}_{i\in A},\phi\big{)}$

satisfies

$A\prec^{\star}B\Rightarrow f(A)<f(B),$

which implies that

$\omega^{\alpha}+1={\sf
rk}\big{(}\mathcal{S}_{\alpha},\prec^{\star}\big{)}\leqslant{\sf
rk}\big{(}{\mathcal{C}oa}\big{(}{\mathbb{Z},(e_{n}),E}\big{)},<\big{)}.$

Therefore, if $\mathbb{Z}[\mathcal{S}_{\alpha}]_{c_{0}}$ coarsely embeds into
$E$ for all $\alpha<\omega_{1}$, we see that the rank ${\sf
rk}\big{(}{\mathcal{C}oa}\big{(}{\mathbb{Z},(e_{n}),E}\big{)},<\big{)}$ is
uncountable and therefore that $c_{0}$ coarsely embeds into $E$. The reverse
implication is immediate. ∎

##  3\. A remark on extracting equi-coarse embeddings

If $A$ and $B$ are two subsets of $\mathbb{N}$, we say that $A$ is _almost
contained_ in $B$, written $A\subseteq^{*}B$, if $|A\setminus B|<\aleph_{0}$.
A infinite set $D\subseteq\mathbb{N}$ is called a _pseudo-intersection_ of a
family $\mathcal{C}$ of subsets of $\mathbb{N}$ provided that
$D\subseteq^{*}C$ for all $C\in\mathcal{C}$. In [BLMS_FM], Proposition 1.1 was
established via the following diagonalization principle.

######  Lemma 3.1 (MA+$\neg$CH).

Let $\\{A_{\alpha}\\}_{\alpha<\omega_{1}}$ be a family of infinite subsets of
$\mathbb{N}$ such that $A_{\beta}\subseteq^{*}A_{\alpha}$ whenever
$\alpha<\beta$. Then $\\{A_{\alpha}\\}_{\alpha<\omega_{1}}$ has a pseudo-
intersection.

Given an ordinal $\lambda$, a _$\lambda$ -tower_ is a family
$\\{A_{\alpha}\\}_{\alpha<\lambda}$ of infinite subsets of $\mathbb{N}$ with
no pseudo-intersection and such that $A_{\beta}\subseteq^{*}A_{\alpha}$
whenever $\alpha<\beta<\lambda$. We then define the cardinal number
$\mathfrak{t}$ by

$\mathfrak{t}=\min\\{|\lambda|\colon\text{there exists a
}\lambda\text{-tower}\\}.$

Whereas it is easy to see that $\aleph_{1}\leqslant\mathfrak{t}\leqslant
2^{\aleph_{0}}$, the specific value of $\mathfrak{t}$ is highly sensitive to
additional set theoretical assumptions. For example, under the assumption of
$\text{MA}_{\sigma\text{-centered}}$ (a slight weakening of MA), we have

$\mathfrak{t}=2^{\aleph_{0}}$

(see [JustWeese, Theorem 19.24, Theorem 19.25]), which immediately implies
Lemma 3.1. Therefore, the argument from [BLMS_FM] could be straightforwardly
extended to other uncountable cardinals strictly less than $2^{\aleph_{0}}$.
Here we present a different way to derive Proposition 1.1 in terms of another
cardinal number, the bounding number $\mathfrak{b}$, that is more naturally
related to the problem at hand.

For functions $f,g\colon\mathbb{N}\to\mathbb{N}$, set

$f<^{*}g\;\Leftrightarrow\;\text{there is some $n$ so that }f(m)<g(m)\text{
for all }m\geqslant n.$

In this case, we say that $g$ eventually dominates $f$. The relation $<^{*}$
defines a partial order on $\mathbb{N}^{\mathbb{N}}$ and we say that a family
$F\subseteq\mathbb{N}^{\mathbb{N}}$ is _bounded_ (with respect to $<^{*}$) if
there exists $g\in\mathbb{N}^{\mathbb{N}}$ such that $f<^{*}g$ for all $f\in
F$. If a family is not bounded it is called an _unbounded_ family. The
_bounding number_ is the cardinal number $\mathfrak{b}$ defined by

$\mathfrak{b}=\min\\{|F|\colon F\subseteq\mathbb{N}^{\mathbb{N}}\;\&\;F\text{
is unbounded}\\}.$

The following inequalities hold without any additional assumptions (see
[JustWeese, Theorem 19.24]),

$\aleph_{1}\leqslant\mathfrak{t}\leqslant\mathfrak{b}\leqslant
2^{\aleph_{0}}.$

In particular, $\text{MA}_{\sigma\text{-centered}}$ and $\neg\,\text{CH}$
imply that $\mathfrak{b}=2^{\aleph_{0}}>\aleph_{1}$.

The next lemma relates the bounding number to the boundedness problem for
families of non-negative maps and will subsequently be applied to the families
of compression and expansion moduli of a family of coarse embeddings.

######  Lemma 3.2 ($\mathfrak{b}>\aleph_{1}$).

Suppose $\\{g_{i}\\}_{i\in I}$ and $\\{f_{i}\\}_{i\in I}$ are uncountable
families of non-decreasing functions $g_{i},f_{i}\in\mathbb{N}^{\mathbb{N}}$
satisfying $\lim_{n\to\infty}g_{i}(n)=\lim_{n\to\infty}f_{i}(n)=\infty$. Then
there are non-decreasing functions $f,g\in\mathbb{N}^{\mathbb{N}}$ also
satisfying $\lim_{n\to\infty}g(n)=\lim_{n\to\infty}f(n)=\infty$ and an
uncountable subset $J\subseteq I$ so that

$g\leqslant g_{i},\quad f_{i}\leqslant f\quad\text{ for all $i\in J$}.$

###### Proof.

Without loss of generality, we have $|I|=\aleph_{1}$. In order to bound the
$g_{i}$ from below we will bound the collection of generalized inverses
$g^{\vee}_{i}$ from above. Because $\lim_{n\to\infty}g_{i}(n)=\infty$, the
generalized inverse $g^{\vee}_{i}\in\mathbb{N}^{\mathbb{N}}$ may be defined by

$g^{\vee}_{i}(k)=\min\\{n\;\big{|}\;g_{i}(n)>k\\}$

and hence satisfies

(1) $g_{i}\big{(}g^{\vee}_{i}(k)\big{)}>k.$

Also, because $|I|<\mathfrak{b}$, the families $F=\\{f_{i}\\}_{i\in I}$ and
$G^{\vee}=\\{g^{\vee}_{i}\\}_{i\in I}$ are bounded and we may therefore choose
some $h\in\mathbb{N}^{\mathbb{N}}$ so that

$g^{\vee}_{i},f_{i}<^{*}h\quad\text{for all $i\in I$}.$

Replacing, if necessary $h$ by the function $h^{\uparrow}$ defined by
$h^{\uparrow}(n)=\max_{m\leqslant n}h(m)$, we may also suppose that $h$ is
non-decreasing. Since clearly $\lim_{n\to\infty}h(n)=\infty$, we can take
$f=h$ as the upper bound for the collection $F$.

Consider the generalized inverse $h^{\vee}$ defined by

$h^{\vee}(k)=\min\\{n\;\big{|}\;h(n)>k\\},$

and note that, for all sufficiently large $k$, we have

(2) $h\big{(}h^{\vee}(k)-1\big{)}\leqslant k.$

Also, $h^{\vee}$ is non-decreasing and $\lim_{n\to\infty}h^{\vee}(n)=\infty$.
Define $g\in\mathbb{N}^{\mathbb{N}}$ by

$g(n)=\max\\{1,h^{\vee}(n)-1\\}.$

Then, given $i\in I$, pick some $p$ so that both

$g^{\vee}_{i}(k)<h(k)$

and (2) hold for all $k\geqslant p$. Then, for all $n\geqslant p$ large enough
so that $g(n)>p$, we have by definition of $h^{\vee}$ that

$g^{\vee}_{i}(g(n))<h\big{(}g(n)\big{)}=h\big{(}h^{\vee}(n)-1\big{)}\stackrel{{\scriptstyle\eqref{eq:h}}}{{\leqslant}}n.$

and so, because $g_{i}$ is non-decreasing,

$g(n)\stackrel{{\scriptstyle\eqref{eq:g}}}{{<}}g_{i}\big{(}g^{\vee}_{i}(g(n)\big{)}\leqslant
g_{i}(n).$

Now, for all $i\in I$, let $n_{i}$ be minimal so that

$g(n)<g_{i}(n),\quad f_{i}(n)<f(n)$

for all $n\geqslant n_{i}$. Since $I$ is uncountable, there is some
uncountable set $J\subseteq I$ and some $m$ so that $n_{i}=m$ for all $i\in
J$. By replacing $f$ and $g$ by the functions

$f^{\prime}(n)=\max\\{f(n),f(m)\\}$

and

$g^{\prime}(n)=\begin{cases}1&\text{ if }n<m,\\\ g(n)&\text{ if }n\geqslant
m,\end{cases}$

we can then ensure that $g\leqslant g_{i}$ and $f_{i}\leqslant f$ for all
$i\in J$. ∎

######  Proposition 3.3 ($\mathfrak{b}>\aleph_{1}$).

Suppose $Y$ is a metric space and $\\{X_{i}\\}_{i\in I}$ is an uncountable
family of metric spaces so that each $X_{i}$ coarsely embeds into $Y$. Then
there is an uncountable subfamily $\\{X_{i}\\}_{i\in J}$ that equi-coarsely
embeds into $Y$, i.e., so that all the $X_{i}$ with $i\in J$ embed into $Y$
with the same moduli $\kappa$ and $\omega$.

###### Proof.

For every $i\in I$, there are maps $f_{i}\colon X_{i}\to Y$ and non-decreasing
functions $\kappa_{i},\omega_{i}\colon[0,\infty)\to[0,\infty)$ so that
$\lim_{t\to\infty}\kappa_{i}(t)=\infty$ and, for all $x_{1},x_{2}\in X_{i}$,

$\kappa_{i}(d_{X_{i}}(x_{1},x_{2}))\leqslant
d_{Y}(f(x_{1}),f_{i}(x_{2}))\leqslant\omega_{i}(d_{X_{i}}(x_{1},x_{2})).$

We apply Lemma 3.2 to the maps $g_{i}(n)=\lfloor\kappa_{i}(n)+1\rfloor$ and
$f_{i}(n)=\lfloor\omega_{i}(n)+1\rfloor$, which gives us non-decreasing
functions $g,f\in\mathbb{N}^{\mathbb{N}}$ so that $g\leqslant g_{i}$ and
$f_{i}\leqslant f$ for an uncountable number of $i$. Setting

$\kappa(t)=g(\lfloor t\rfloor)-1,\qquad\omega(t)=f(\lceil t\rceil)$

for all $t\geqslant 1$ and $\kappa(t)=0$, $\omega(t)=f(1)$ for $t\in[0,1)$, we
see that $\kappa\leqslant\kappa_{i}$ and $\omega_{i}\leqslant\omega$ for an
uncountable set of $i$’s. ∎

## References