Lower-bounding S(N) beyond N/3 by an unbounded function

Establish the existence of a function ω(N) → ∞ such that for every positive integer N, S(N) ≥ N/3 + ω(N), where S(N) = min_{A ⊂ ℕ, |A| = N} S(A) and S(A) denotes the maximum size of a sum-free subset of A (a subset containing no x, y, z with x + y = z).

Background

Let S(N) denote the minimum, over all sets A of N positive integers, of the size of the largest sum-free subset of A. Erdős proved the classical lower bound S(N) ≥ N/3. Subsequent improvements by Alon–Kleitman and by Bourgain raised the bound to (N+1)/3 and (N+2)/3, respectively, but no asymptotically unbounded improvement over N/3 had been established.

A widely believed strengthening asked whether one can improve the Erdős–Alon/Kleitman–Bourgain lower bound by an arbitrarily large amount, i.e., whether S(N) ≥ N/3 + ω(N) for some ω(N) → ∞. This question was a longstanding open problem and appeared as Problem 1 on Green’s list of 100 open problems. The present paper proves S(N) ≥ N/3 + c log log N, thereby answering this in the affirmative.