The sum-free process

TL;DR

This paper analyzes a probabilistic process for constructing large sum-free subsets in cyclic groups, establishing lower bounds on their size and revealing non-pseudorandom structural properties.

Contribution

It provides a new lower bound on the size of sum-free sets generated by a random greedy process, aligning with existing threshold results and highlighting unique structural features.

Findings

Lower bound on sum-free set size with high probability

Alignment with known threshold results

Identification of non-pseudorandom properties

Abstract

$S \subseteq \mathbb{Z}_{2n}$ is said to be sum-free if $S$ has no solution to the equation $a+b=c$. The sum-free process on $\mathbb{Z}_{2n}$ starts with $S:=\emptyset$, and iteratively inserts elements of $\mathbb{Z}_{2n}$, where each inserted element is chosen uniformly at random from the set of all elements that could be inserted while maintaining that $S$ is sum-free. We prove a lower bound (which holds with high probability) on the final size of $S$, which matches a more general result of Bennett and Bohman, and also matches the order of a sharp threshold result proved by Balogh, Morris and Samotij. We also show that the set $S$ produced by the process has a particular non-pseudorandom property, which is in contrast with several known results about the random greedy independent set process on hypergraphs.