Sum-free sets in groups: a survey

We discuss several questions concerning sum-free sets in groups, raised by Erdős 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-avoiding with respect to $A$). In the above mentioned survey, Erdős conjectured that if $\lvert A \rvert$ 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 $k \geq 4$. On the other hand, using the new characterization result, we are able to prove a positive result in the case when $\lvert G \rvert$ is not divisible by small primes.

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T. Tao is supported by NSF grant DMS-0649473 and by a Simons Investigator Award.

V. Vu is supported by research grants DMS-0901216 and AFOSAR-FA-9550-09-1-0167.

Received 16 May 2017

Published 21 June 2017