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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). 

In a recent article, Alon, Hanneke, Holzman, and Moran (FOCS '21) introduced a unifying framework to study the learnability of classes of partial concepts. One of the central questions studied in their work is whether the learnability of a partial concept class is always inherited from the learnability of some "extension" of it to a total concept class. They showed this is not the case for PAC learning but left the problem open for the stronger notion of online learnability. We resolve this problem by constructing a class of partial concepts that is online learnable, but no extension of it to a class of total concepts is online learnable (or even PAC learnable).