Sumfree sets in groups: a survey

TL;DR

This survey explores sum-free sets in groups, providing characterizations for large sets, counterexamples for certain conjectures, and conditions under which positive results hold, advancing understanding of additive combinatorics in group structures.

Contribution

The paper offers a new characterization of large sum-free sets in abelian groups, disproves a conjecture for all k ≥ 4, and establishes conditions for positive results based on group order.

Findings

Counterexamples for all k ≥ 4 to Erdős's conjecture

Characterization of large sum-free sets in abelian groups

Positive results when group order is not divisible by small primes

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.