Sparse Graph Counting and Kelley-Meka Bounds for Binary Systems
Bibliographic record
Abstract
In a recent breakthrough, Kelley and Meka (FOCS 2023) obtained a strong upper bound on the density of sets of integers without non-trivial three-term arithmetic progressions. In this work, we extend their result, establishing similar bounds for all linear patterns defined by binary systems of linear forms, where “binary” indicates that every linear form depends on exactly two variables. Prior to our work, no strong bounds were known for such systems even in the finite field model setting. A key ingredient in our proof is a graph counting lemma. The classical graph counting lemma, developed by Thomason (Random Graphs 1985) and Chung, Graham, and Wilson (Combinatorica 1989), is a fundamental tool in combinatorics. For a fixed graph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H$</tex>, it states that the number of copies of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H$</tex> in a pseudorandom graph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> is similar to the number of copies of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H$</tex> in a purely random graph with the same edge density as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex>. However, this lemma is only non-trivial when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> is a dense graph. In this work, we prove a graph counting lemma that is also effective when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> is sparse. Moreover, our lemma is well-suited for density increment arguments in additive number theory. As an immediate application, we obtain a strong bound for the Turán problem in abelian Cayley sum graphs: let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Gamma$</tex> be a finite abelian group with odd order. If a Cayley sum graph on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Gamma$</tex> does not contain any r-elique as a sub graph, it must have at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2^{-\Omega_r\left(\log ^{1 / 16}\vert \Gamma\vert \right)} \cdot\vert \Gamma\vert ^2$</tex> edges. These results hinge on the technology developed by Kelley and Meka and the follow-up work by Kelley, Lovett, and Meka (STOC 2024).
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".