Sets of integers with no large sum-free subset

Abstract

Answering a question of P. Erdős from 1965, we show that for every $\varepsilon> 0$ there is a set $A$ of $n$ integers with the following property: every set $A’ \subset A$ with at least $\left(\frac{1}{3} + \varepsilon\right) n$ elements contains three distinct elements $x,y,z$ with $x + y = z$.