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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 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-elique as a sub graph, it must have at most 2−Ωr(log1/16|Γ|)⋅|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). 

We construct an explicit and structured family of 3XOR instances which is hard for O(√{log n}) levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems are highly structured and can be constructed explicitly in deterministic polynomial time. Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov. Our construction offers an interesting contrast to the recent work of Alev, Jeronimo and the last author (FOCS 2019). They showed that 3XOR instances in which the variables correspond to vertices in a high-dimensional expander are easy to solve. In contrast, in our instances the variables correspond to the edges of the complex.