Datasets:

CohenQu
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TCS-MATH-v2

problem stringlengths 0 6.4k	answer stringlengths 0 19.2k	rubrics stringlengths 0 4.07k	source stringclasses 20 values	difficulty stringclasses 3 values
Let - \([k]=\{1,2,\dots,k\}\), - \([k]^d\) be the \(d\)-dimensional grid, - a combinatorial line in \([k]^d\) mean a set of \(k\) points obtained by fixing some coordinates and letting all remaining coordinates vary synchronously through \(1,2,\dots,k\), and - an \(\varepsilon\)-net for a finite point set \(X\...	{"visible_target": "For every sufficiently large constant C>0, there exist n, ε>0, and a set X of n points in the plane such that every ε-net for lines for X has size larger than C/ε.", "intermediate_lemmas": [{"node_id": "node_19", "statement": "For every positive integer d there exist planar vectors v_1,\\dots,v_d su...	Total: 7 points 1. [2 pt] Identify the projection/vector bottleneck Names the planar-vector lemma (or an equivalent faithful summary of node_19) and explains that it preserves direction information for small integer combinations, preventing accidental extra collinearities in the projection. Partial credit if the an...	FOCS_2010	hard
Let [k]={1,2,...,k}, and let [k]^d be the d-dimensional grid over [k]. A combinatorial line in [k]^d is a set of k points obtained by fixing some coordinates and letting all remaining coordinates vary together through the values 1,2,...,k. For a finite planar point set X, a subset N⊂X is an ε-net for lines if every ge...	{"visible_target": "For every sufficiently large constant C, there exist n, ε>0, and a set X of n points in the plane such that every ε-net for lines for X has size greater than C/ε.", "intermediate_lemmas": [{"node_id": "node_19", "statement": "For every positive integer d there exist planar vectors v1,...,vd such tha...	Total: 7 points 1. [2 pt] Vector-faithfulness lemma Identifies the planar vector lemma (or an equivalent DAG-grounded formulation) as a central bottleneck, and explains that it prevents accidental directional/collinearity collisions when projecting [k]^d into the plane. 2. [2 pt] Combinatorial-line to geometric-lin...	FOCS_2010	medium
Let \([k]=\{1,2,\dots,k\}\), and let \([k]^d\) denote the \(d\)-dimensional grid. \nA combinatorial line in \([k]^d\) is a set of \(k\) grid points obtained by fixing some coordinates and letting all remaining coordinates vary together through the values \(1,2,\dots,k\). \nFor a finite point set \(A\) in the plane, a s...	{"visible_target": "For every sufficiently large constant C>0, there exist n, ε>0, and a planar n-point set X such that every ε-net for line ranges for X has size greater than C/ε.", "intermediate_lemmas": [{"node_id": "node_19", "statement": "There exist planar vectors v_1,\\dots,v_d such that for integer coefficient ...	Total: 7 points 1. [2 pt] Density Hales-Jewett as the combinatorial engine Identifies the dense-subset-implies-combinatorial-line result (node_18, or an equivalent faithful description) and explains that it is the source of unavoidable line structure in large subsets, later transferred to the plane. 2. [2 pt] Direc...	FOCS_2010	hard
Let an ε-net for a finite planar point set X with respect to line ranges mean a subset N ⊆ X that intersects every line containing at least ε\|X\| points of X. Use the following notation. - For an integer k ≥ 2, write [k] = {1,2,…,k} and [k]^d for the d-dimensional grid. - A combinatorial line in [k]^d is a set of k gri...	{"visible_target": "For every sufficiently large constant C, there exist n, ε > 0, and a finite set X of n points in the plane such that every ε-net for line ranges on X has size greater than C/ε.", "intermediate_lemmas": [{"node_id": "node_19", "statement": "For every positive integer d there exist planar vectors v1,…...	Total: 7 points 1. [2 pt] Identify the combinatorial engine Correctly identifies Density Hales-Jewett / Fact 1 (node_18) as a major lemma and explains that it forces a combinatorial line inside every sufficiently dense subset of [k]^d, specifically at density 1/2, which is then used to obstruct small ε-nets. 2. [2 ...	FOCS_2010	hard
Consider the following target theorem. Target theorem. For every sufficiently large constant C, there exists a sequence of positive reals ε_i tending to 0 such that, for every ε = ε_i in the sequence and for all sufficiently large n, there is a set Y_n of n points in general position in the plane for which every stron...	{"visible_target": "For every sufficiently large constant C, there exists a sequence ε_i → 0 such that for each ε = ε_i and all sufficiently large n, there is an n-point planar set Y_n in general position whose smallest strong ε-net for fat lines has size greater than C/ε.", "intermediate_lemmas": [{"node_id": "node_09...	Total: 7 points 1. [2 pt] Identifies the imported line-range lower bound The answer identifies the previously proved strong ε-net lower bound for lines on X (node_09, or a clearly equivalent DAG-grounded formulation) and explains that Theorem 1.2 reuses it as the main engine rather than reproving everything. Good e...	FOCS_2010	hard
Let a strong ε-net for a finite planar point set A with respect to a family of ranges be a subset N ⊆ A that intersects every range containing at least an ε-fraction of the points of A. A fat line in the plane means the set of all points within some fixed distance μ from a line. Consider the following target theorem:...	{"visible_target": "For every sufficiently large constant C, there exists a sequence of positive reals ε_i → 0 such that for every ε = ε_i in the sequence and for every sufficiently large n, there is a set Y_n of n points in general position in the plane for which every strong ε-net with respect to fat lines has size g...	Total: 7 points 1. [2 pt] Identify prior line-range lower bound as the main inherited bottleneck The answer should identify the previous strong ε-net lower bound for lines on X (node_09, or a faithful equivalent DAG-grounded formulation) as a central ingredient, and explain that Theorem 1.2 reuses it to force a mis...	FOCS_2010	hard
Let - [k] = {1,2,...,k}, and [k]^d be the d-dimensional grid; - a combinatorial line in [k]^d mean a set of k points obtained by fixing some coordinates and letting all remaining coordinates vary synchronously through 1,2,...,k; - a weak ε-net for a finite planar point set X with respect to lines mean a set Y of point...	{"visible_target": "For every sufficiently large constant C, there exist n and ε > 0 and a set X of n points in the plane such that every weak ε-net for X with respect to lines has size greater than C/ε.", "intermediate_lemmas": [{"node_id": "node_29", "statement": "In R^d, if a geometric line is called special when it...	Total: 7 points 1. [2 pt] Intersection bottleneck in high dimension Identifies the special-line intersection claim (only grid points lie on two or more special lines in R^d, node_29) and explains that it is the new ingredient needed for weak nets because off-X points must have limited hitting power. 2. [2 pt] Proje...	FOCS_2010	hard
Let - \([k]=\{1,2,\dots,k\}\), - \([k]^d\) be the \(d\)-dimensional grid, - a combinatorial line in \([k]^d\) mean a set of \(k\) grid points obtained by fixing some coordinates and letting all remaining coordinates vary synchronously through \(1,2,\dots,k\), - a weak \(\varepsilon\)-net for a finite point set \(X\sub...	{"visible_target": "For every sufficiently large constant C>0, there exist n, ε>0, and a finite set X⊂R^2 such that every weak ε-net for X with respect to lines has size greater than C/ε.", "intermediate_lemmas": [{"node_id": "node_29", "statement": "If two distinct special lines in R^d intersect, then their intersecti...	Total: 7 points 1. [2 pt] Identify the intersection bottleneck Correctly identifies the claim that distinct special lines intersect only at grid points (node_29), and explains that this controls how many special lines a non-grid / non-X point can stab, which is the key extra issue in the weak-net setting. 2. [2 pt]...	FOCS_2010	hard
Let a weak ε-net for a finite point set X in the plane with respect to lines mean a set Y of points in the plane, not necessarily contained in X, such that every line containing at least ε\|X\| points of X contains at least one point of Y. Consider the following target theorem. Target theorem. For every sufficiently la...	{"visible_target": "For every sufficiently large constant C, there exist n, ε > 0, and a set X of n points in the plane such that every weak ε-net for X with respect to lines has size greater than C/ε.", "intermediate_lemmas": [{"node_id": "node_29", "statement": "In R^d, the only points belonging to at least two disti...	Total: 7 points 1. [2 pt] Identify the higher-dimensional intersection lemma The answer identifies the claim that distinct special lines in R^d can intersect only at grid points [k]^d, and explains that this controls how much a weak-net point outside the original set can help. Partial credit if the intersection-con...	FOCS_2010	hard
Let a weak ε-net for a finite point set X in the plane with respect to lines mean a set Y of points in the plane, not necessarily contained in X, such that every line containing at least ε\|X\| points of X meets Y. Consider the following target theorem: Target theorem. For every sufficiently large constant C, there exi...	{"visible_target": "For every sufficiently large constant C, there exist n, ε > 0, and a set X of n points in the plane such that every weak ε-net for X with respect to lines has size greater than C/ε.", "intermediate_lemmas": [{"node_id": "node_29", "statement": "Intersections of special lines in [k]^d occur only at g...	Total: 7 points 1. [2 pt] Identify the special-line intersection bottleneck The answer identifies the claim that distinct special lines in R^d intersect only at grid points (node_29), and explains that this is the new weak-net-specific ingredient controlling how many special lines an off-set point can hit. 2. [1 pt...	FOCS_2010	hard
Consider the following target lemma. Target lemma. Fix an integer k \(\ge 2\). For every positive integer \(d\), there exist vectors \(v_1,\dots,v_d\in \mathbb R^2\) such that for every two nonzero integer vectors \((k_1,\dots,k_d)\) and \((k'_1,\dots,k'_d)\) with \(\|k_i\|,\|k'_i\|<k\) for all \(i\), the planar vectors \...	{"visible_target": "Fix an integer k \\(\\ge 2\\). For every positive integer \\(d\\), there exist vectors \\(v_1,\\dots,v_d\\in \\mathbb R^2\\) such that for every two nonzero integer vectors \\((k_1,\\dots,k_d)\\) and \\((k'_1,\\dots,k'_d)\\) with \\(\|k_i\|,\|k'_i\|<k\\), the sums \\(\\sum_i k_i v_i\\) and \\(\\sum_i k'...	Total: 7 points 1. [4 pt] Identify the probabilistic-construction bottleneck The answer identifies the random-coordinate/union-bound strategy from the DAG (node_20, or an equivalently faithful description tied to the actual proof structure), and explains that the target lemma is proved by encoding unwanted same-dir...	FOCS_2010	medium
Let k be a fixed integer at least 2. Consider the following target lemma. Target lemma. For every positive integer d there exist vectors v_1,\dots,v_d in the plane such that for every two nonzero integer coefficient vectors a=(a_1,\dots,a_d) and b=(b_1,\dots,b_d) with \|a_i\|,\|b_i\|<k for all i, the planar vectors \[ \su...	{"visible_target": "For every positive integer d there exist vectors v_1,\\dots,v_d in the plane such that for every two nontrivial integer sequences (k_1,\\dots,k_d) and (k'_1,\\dots,k'_d), with \|k_i\|,\|k'_i\|<k for all i, the two vectors \\sum_i k_i v_i and \\sum_i k'_i v_i have the same direction iff the coefficient v...	Total: 7 points 1. [2 pt] Identifies Schwartz-Zippel as a key ingredient The answer identifies Fact 1 / Schwartz-Zippel as an essential intermediate result and explains that it bounds the probability that a bad directional-collision polynomial vanishes. Partial credit if it mentions probabilistic polynomial vanishi...	FOCS_2010	medium
Let [k]={1,2,...,k} and let [k]^d be the d-dimensional grid. A combinatorial line in [k]^d is a set of k points obtained by fixing some coordinates and letting all remaining coordinates vary synchronously through 1,2,...,k. For a finite point set A in the plane, a subset N⊂A is an ε-net for lines if every line contai...	{"visible_target": "Any subset of X of size at least \|X\|/2 contains all k points of some special line of X; consequently, its complement is not an ε-net for lines for X.", "intermediate_lemmas": [{"node_id": "node_18", "statement": "Density Hales-Jewett theorem: for fixed k and δ>0, there exists d_0(k,δ) such that for ...	Total: 7 points 1. [3 pt] Identifies density Hales-Jewett as the main forcing lemma The answer should identify the density theorem on [k]^d as a central intermediate result and explain that it is applied to the preimage of a half-sized subset of X to force a combinatorial line. Full credit requires both identificat...	FOCS_2010	medium
Consider the following setup. Let \(X\) be a planar point set obtained from the grid \([k]^d\) by a construction with the property that every combinatorial line in \([k]^d\) corresponds to a geometric line in the plane containing exactly the associated \(k\) points of \(X\) and no other points of \(X\). A combinatori...	{"visible_target": "Any subset of fewer than \|X\|/2 points of Y_n must completely miss at least half of the clusters S_x, and therefore misses an entire fat line corresponding to one combinatorial line.", "intermediate_lemmas": [{"node_id": "node_26", "statement": "Blow up each point x in X to a nearby cluster S_x, chos...	Total: 7 points 1. [2 pt] Cluster blow-up to fat lines Identifies the construction replacing each x in X by a cluster S_x and explains that it is what turns a missed line-worth of original points into a missed fat line in Y_n. Full credit requires explaining the no-other-points property of the resulting fat line. 2...	FOCS_2010	medium
Let [k] = {1,2,...,k}, and let [k]^d be the d-dimensional grid. A combinatorial line in [k]^d is a set of k points obtained by fixing some coordinates and letting all remaining coordinates vary together through 1,2,...,k. A geometric line in R^d is called special if it contains all k points of some combinatorial line....	{"visible_target": "With probability 1, every special planar line contains exactly k points of X, and the only points in the plane that lie on more than two special planar lines are the points of X.", "intermediate_lemmas": [{"node_id": "node_29", "statement": "Intersections of special lines in R^d occur only at grid p...	Total: 7 points 1. [3 pt] Identify the high-dimensional intersection lemma The answer identifies the claim that only points of [k]^d lie on at least two distinct special lines in R^d, and explains that this is the central source of the planar bounded-multiplicity conclusion after random projection. Partial credit i...	FOCS_2010	hard
Let X be a finite set of points in the plane, and call N ⊆ X an ε-net for line ranges if every line containing at least ε\|X\| points of X meets N. A combinatorial line in [k]^d is a set of k points obtained by fixing some coordinates and letting the remaining coordinates vary together through 1,2,...,k. You may use th...	{"main_obstacle": "The key difficulty is not the final counting step, but constructing a planar point set whose collinearity pattern faithfully simulates combinatorial lines in [k]^d. One needs many guaranteed k-point lines coming from the combinatorial structure, while preventing accidental additional points of X from...	Total: 7 points 1. [2 pt] Main obstacle identified Recognizes that the core issue is faithfully converting combinatorial lines in [k]^d into planar lines while avoiding unintended collinearities, not merely doing the final counting. 2. [1 pt] Meaningful construction of X Proposes a concrete encoding of [k]^d int...	FOCS_2010	hard
Let X be a finite set of points in the plane, and say that a subset N \subseteq X is a strong \epsilon-net for lines if every line containing at least \epsilon \|X\| points of X intersects N. A combinatorial line in [k]^d is a set of k points obtained by fixing some coordinates and letting all remaining coordinates vary...	{"main_obstacle": "The key obstacle is to transfer the rich line structure guaranteed combinatorially in the high-dimensional grid [k]^d into actual planar lines, while preventing spurious collinearities. Density Hales-Jewett only says that dense subsets of [k]^d contain combinatorial lines; to turn this into an epsilo...	Total: 8 points 1. [2 pt] Main obstacle identified Clearly identifies that the bottleneck is not DHJ itself, but faithfully embedding combinatorial lines from [k]^d as planar lines while avoiding unintended extra collinearities/parallelism that would break the argument. 2. [1 pt] Meaningful construction of X Pro...	FOCS_2010	hard
Let an \(\varepsilon\)-net for a finite planar point set \(A\) with respect to a family of ranges mean a subset \(N\subseteq A\) that intersects every range containing at least an \(\varepsilon\)-fraction of \(A\). A fat line is the set of all points within distance \(\mu\) of some line. Assume the following backgroun...	{"main_obstacle": "The base construction only gives one specific size \\(\|X\|=k^d\\), with many exact collinearities, and the heavy ranges are ordinary lines containing exactly \\(k\\) selected points. To prove the stronger theorem one must simultaneously: (i) allow arbitrary larger \\(n\\), not just \\(n=\|X\|\\); (ii) d...	Total: 7 points 1. [1 pt] Identifies the main obstacle Recognizes that the challenge is not the original line lower bound itself, but extending it to arbitrary larger n and general position while preserving heavy ranges via fat lines and avoiding accidental extra incidences. 2. [2 pt] Proposes the cluster blow-up c...	FOCS_2010	medium

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TCS-MATH-v2

A merged view of all data files from AI-Math-TCS/tcs_dataset_v2 at source revision 4c073da586707d094401e03a99d4fda93e43112e.

Schema

The train split contains 73,390 rows with exactly these string columns:

problem
answer
rubrics
source
difficulty

The source dataset's rubric column was renamed to rubrics. The source value is the source configuration/conference-year (for example, FOCS_2024). All 20 source Parquet files and all source rows were retained without deduplication or filtering.

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