# Symmetric Complete Sum-free Sets in Cyclic Groups

## TL;DR

This paper constructs symmetric complete sum-free sets in cyclic groups, demonstrating their density and exponential quantity, and fully characterizing large such sets in prime cyclic groups.

## Contribution

It introduces new constructions of symmetric complete sum-free sets in cyclic groups and characterizes large sets in prime cyclic groups, answering longstanding questions.

## Key findings

- Relative sizes are dense in [0, 1/3]
- Number of sets is exponential in group order
- Complete characterization for large sets in prime groups

## Abstract

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.