Let f (n, v, e) denote the maximum number of edges in a 3-uniform hypergraph not containing e edges spanned by at most v vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges e ≥ 3, what is the smallest integer d = d(e) so that f (n, e + d, e) = o(n 2 )? This question has its origins in work of Brown, Erdős and Sós from the early 70's and the standard conjecture is that d(e) = 3 for every e ≥ 3. The state of the art result regarding this problem was obtained in 2004 by Sárközy and Selkow, who showed that f (n, e + 2 + ⌊log 2 e⌋, e) = o(n 2 ). The only improvement over this result was a recent breakthrough of Solymosi and Solymosi, who improved the bound for d(10) from 5 to 4. We obtain the first asymptotic improvement over the Sárközy-Selkow bound, showing that f (n, e + O(log e/ log log e), e) = o(n 2 ).

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We consider the problem of counting the number of copies of a fixed graph H within an input graph G . This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input G has bounded degeneracy . This is a rich family of graphs, containing all graphs without a fixed minor (e.g. planar graphs), as well as graphs generated by various random processes (e.g. preferential attachment graphs). We say that H is easy if there is a linear-time algorithm for counting the number of copies of H in an input G of bounded degeneracy. A seminal result of Chiba and Nishizeki from ’85 states that every H on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all H on 5 vertices, and further proved that for every k > 5 there is a k -vertex H which is not easy. They left open the natural problem of characterizing all easy graphs H . Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph H to be easy. Here we show that this sufficient condition is also necessary, thus fully answering the Bera–Pashanasangi–Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms.

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