Sets of integers with no large sum-free subset

TL;DR

This paper constructs large sets of integers where any sufficiently large subset necessarily contains a solution to x + y = z, addressing a question posed by P. Erdos in 1965.

Contribution

It provides explicit constructions of integer sets with no large sum-free subsets, answering a longstanding open problem.

Findings

For every epsilon > 0, there exists a set of n integers with the property that large subsets contain solutions to x + y = z.

The result confirms the existence of sets with no large sum-free subsets for any positive epsilon.

Addresses a question by P. Erdos from 1965 about the structure of sum-free subsets.

Abstract

Answering a question of P. Erdos from 1965, we show that for every eps>0 there is a set A of n integers with the following property: every subset A' of A with at least (1/3 + eps)n elements contains three distinct elements x,y,z with x + y = z.