Sparse graph counting and Kelley-Meka bounds for binary systems

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 translation-invariant 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 a sample application, we obtain a strong bound for the Turán problem in Abelian Cayley graphs: we prove that an Abelian Cayley graph on N vertices that does not contain any r-clique as a subgraph must have at most 2^-Ω_r(log^1/16N)· N^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).