On the number of sum-free triplets of sets

TL;DR

This paper counts the number of sum-free triplets of subsets in cyclic groups, providing improved estimates and extending results to general abelian groups using a novel, simplified approach.

Contribution

It introduces a new, simpler method to estimate the number of sum-free triplets, improving previous bounds and extending results to broader abelian groups.

Findings

Improved bounds on the number of sum-free triplets in cyclic groups

Explicit estimates for smaller order terms

Extension of results to general abelian groups

Abstract

We count the ordered sum-free triplets of subsets in the group $\mathbb{Z}/p\mathbb{Z}$, i.e., the triplets $(A,B,C)$ of sets $A,B,C \subset \mathbb{Z}/p\mathbb{Z}$ for which the equation $a+b=c$ has no solution with $a\in A$, $b \in B$ and $c \in C$. Our main theorem improves on a recent result by Semchankau, Shabanov, and Shkredov using a different and simpler method. Our proof relates previous results on the number of independent sets of regular graphs by Kahn, Perarnau and Perkins, and Csikv\'ari to produce explicit estimates on smaller order terms. We also obtain estimates for the number of sum-free triplets of subsets in a general abelian group.