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Sparse graph counting and Kelley-Meka bounds for binary systems

Yuval Filmus, Hamed Hatami, Kaave Hosseini, Esty Kelman

TL;DR

This paper develops a sparse graph counting lemma that remains effective in sparse graphs and uses it to extend Kelley–Meka bounds from 3-term progressions to all binary systems of linear forms. Central to the approach are grid semi-norms, Bohr-set machinery, and density-increment strategies on rectangles and Bohr sets, which yield strong bounds for the number of non-degenerate translations of binary systems in finite abelian groups. The authors prove a Kelley–Meka-type bound: for a binary system $\mathcal{L}$ over a finite abelian group $G$ with $(|G|,\text{coefficients})=1$, any $L$-free set $A$ has size $|A| \le |G|\cdot 2^{-\Omega_{\mathcal{L}}(\log^{1/16}|G|)}$, with improvement to $2^{-\Omega_{\log^{1/9}|G|}}$ when the underlying graph is $2$-degenerate. They also derive a Turán-type bound for Cayley sum graphs over abelian groups, giving $2^{-\,\Omega_r(\log^{1/16}|\Gamma|)}$ edge-density for graphs free of $K_r$, under suitable parity constraints. The results advance sparse analogs of the graph counting paradigm and illuminate the limitations and potentials of density-increment methods for broader classes of linear patterns.

Abstract

In a recent breakthrough, Kelley and Meka (FOCS 2023) obtained a strong upper bound on the density of sets of integers without nontrivial 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 $H$, it states that the number of copies of $H$ in a pseudorandom graph $G$ is similar to the number of copies of $H$ in a purely random graph with the same edge density as $G$. However, this lemma is only non-trivial when $G$ is a dense graph. In this work, we prove a graph counting lemma that is also effective when $G$ 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 $Γ$ be a finite abelian group with odd order. If a Cayley sum graph on $Γ$ does not contain any $r$-clique as a subgraph, it must have at most $2^{-Ω_r(\log^{1/16}|Γ|)}\cdot |Γ|^2$ edges. These results hinge on the technology developed by Kelley and Meka and the follow-up work by Kelley, Lovett, and Meka (STOC 2024).

Sparse graph counting and Kelley-Meka bounds for binary systems

TL;DR

This paper develops a sparse graph counting lemma that remains effective in sparse graphs and uses it to extend Kelley–Meka bounds from 3-term progressions to all binary systems of linear forms. Central to the approach are grid semi-norms, Bohr-set machinery, and density-increment strategies on rectangles and Bohr sets, which yield strong bounds for the number of non-degenerate translations of binary systems in finite abelian groups. The authors prove a Kelley–Meka-type bound: for a binary system over a finite abelian group with , any -free set has size , with improvement to when the underlying graph is -degenerate. They also derive a Turán-type bound for Cayley sum graphs over abelian groups, giving edge-density for graphs free of , under suitable parity constraints. The results advance sparse analogs of the graph counting paradigm and illuminate the limitations and potentials of density-increment methods for broader classes of linear patterns.

Abstract

In a recent breakthrough, Kelley and Meka (FOCS 2023) obtained a strong upper bound on the density of sets of integers without nontrivial 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 , it states that the number of copies of in a pseudorandom graph is similar to the number of copies of in a purely random graph with the same edge density as . However, this lemma is only non-trivial when is a dense graph. In this work, we prove a graph counting lemma that is also effective when 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 be a finite abelian group with odd order. If a Cayley sum graph on does not contain any -clique as a subgraph, it must have at most edges. These results hinge on the technology developed by Kelley and Meka and the follow-up work by Kelley, Lovett, and Meka (STOC 2024).
Paper Structure (32 sections, 26 theorems, 202 equations)

This paper contains 32 sections, 26 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. The sparse graph counting lemma
  3. A density increment counting lemma.
  4. Kelley--Meka bounds for binary systems of linear forms
  5. Turán's theorem for Cayley sum graphs over abelian groups
  6. Acknowledgment
  7. Layout of the paper.
  8. Notation and terminology
  9. Systems of linear forms.
  10. Translation-invariance.
  11. Graph counting lemma
  12. Proof of Theorem \ref{['thm:low_disc_graph_count']}
  13. Proof of Theorem \ref{['thm:main_graph_count']}
  14. Grid norms.
  15. Main technical lemma.
  16. ...and 17 more sections

Key Result

Theorem 1.1

For every fixed graph $H$ with $m$ edges and $\varepsilon \in (0,1)$, there exists $\delta=\delta(\varepsilon,m)>0$ such that the following is true. If $A$ is a graph on $\mathcal{X}$ with density $\alpha$ that satisfies there must exist $S,T \subseteq \mathcal{X}$ with densities $\frac{|S|}{|\mathcal{X}|}, \frac{|T|}{|\mathcal{X}|} \ge \Omega_{m,\varepsilon}(1)$ such that

Theorems & Definitions (58)

  • Theorem 1.1: Graph Counting Lemma MR0930498CGW89
  • Example 1.2
  • Theorem 1.3
  • Theorem 1.4: Main graph counting lemma
  • Remark 1
  • Theorem 1.5: Kelley--Meka-type bounds over finite abelian groups
  • Remark 2
  • Remark 3
  • Corollary 1.6
  • proof
  • ...and 48 more