Symmetric Complete Sum-free Sets in Cyclic Groups

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in $[0,\frac{1}{3}]$, answering a question of Cameron, and that the number of those contained in the cyclic group of order $n$ is exponential in $n$. For primes $p$, we provide a full characterization of the symmetric complete sum-free subsets of ${\mathbb{Z}}_p$ of size at least $(\frac{1}{3}-c)\cdot p$, where $c>0$ is a universal constant.