A note on the largest sum‐free sets of integers

Abstract

AbstractGiven a set of positive integers, an old question in additive combinatorics asks that whether contains a sum‐free subset of size at least for some increasing unbounded function . The question is generally attacked in the literature by considering another conjecture, which asserts that as , . This conjecture, if true, would also imply that a similar phenomenon occurs for ‐sum‐free sets for every . In this note, we prove the latter result directly. The new ingredient of our proof is a structural analysis on the host set , which might be of independent interest.