Power Saving for the Brown-erdos-sos Problem

Let f (n, v, e) denote the maximum number of edges in a 3-uniform hypergraph on n vertices which does not contain v vertices spanning at least e edges. A central problem in extremal combinatorics, famously posed by Brown, Erdos and sos in 1973, asks whether f (n, e + 3, e) = o(n())(2) for every e >= 3. A classical result of Sarkozy and Selkow states that f (n, e + [log(2) e] + 2, e) = o(n(2)) for every e >= 3. This result was recently improved by Motivated by applications to other problems, Gowers and Long made the striking conjecture that f (n, e + 4, e) = O(n(2-epsilon)) for some epsilon = epsilon(e) >0. Conlon, Gishboliner, Levanzov and Shapira, and later, Shapira and Tyomkyn reiterated the following approximate version of this problem. What is the smallest d(e) for which f (n, e + d(e), e) = O(n(2-epsilon)) for some epsilon = epsilon(e) >0? In this paper, we prove that for each e >= 3 we have f (n, e+ [log(2) e] +38,e) = O(n(2-epsilon)) for some epsilon > 0. This shows that one can already obtain power saving near the Sarkozy-Selkow bound at the cost of a small additive constant.