The number of maximal sum-free subsets of integers

TL;DR

This paper establishes an upper bound on the number of maximal sum-free subsets of the set {1, ..., n}, matching the known lower bound asymptotically, using advanced combinatorial tools.

Contribution

It proves that the number of maximal sum-free subsets in {1, ..., n} is at most 2^{(1/4+o(1))n}, confirming a conjecture about their asymptotic growth.

Findings

Number of maximal sum-free subsets is at most 2^{(1/4+o(1))n}

Matches the known lower bound asymptotically

Uses container and removal lemmas of Green

Abstract

Cameron and Erd\H{o}s raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal sum-free sets in $\{1, \dots , n\}$. Our proof makes use of container and removal lemmas of Green as well as a result of Deshouillers, Freiman, S\'os and Temkin on the structure of sum-free sets.