Sumfree sets in groups: a survey

Abstract

We discuss several questions concerning sum-free sets in groups, raised by Erd\H{o}s in his survey "Extremal problems in number theory" (Proceedings of the Symp. Pure Math. VIII AMS) published in 1965. Among other things, we give a characterization for large sets $A$ in an abelian group $G$ which do not contain a subset $B$ of fixed size $k$ such that the sum of any two different elements of $B$ do not belong to $A$ (in other words, $B$ is sum-free with respect to $A$). Erd\H{o}s, in the above mentioned survey, conjectured that if $|A|$ is sufficiently large compared to $k$, then $A$ contains two elements that add up to zero. This is known to be true for $k \leq 3$. We give counterexamples for all $k \ge 4$. On the other hand, using the new characterization result, we are able to prove a positive result in the case when $|G|$ is not divisible by small primes.