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# Journal of Combinatorics

## Volume 8 (2017)

### Number 3

Guest Editor: Steve Butler (Iowa State University)

### Sum-free sets in groups: a survey

Pages: 541 – 552

DOI: http://dx.doi.org/10.4310/JOC.2017.v8.n3.a7

#### Authors

#### Abstract

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.

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.

Paper received on 16 May 2017.