Sharp bound on the number of maximal sum-free subsets of integers

TL;DR

This paper establishes tight bounds on the number of maximal sum-free subsets in the set of integers from 1 to n, confirming they are approximately proportional to 2^{n/4} with precise constants depending on n mod 4.

Contribution

The paper proves sharp asymptotic bounds for the count of maximal sum-free subsets, improving previous lower bounds and using advanced combinatorial and number-theoretic techniques.

Findings

Number of maximal sum-free subsets is approximately C_i * 2^{n/4} for n ≡ i mod 4.

Established constants C_i for each residue class mod 4.

Used container and removal lemmas, structural results, and bounds on small sumset subsets.

Abstract

Cameron and Erd\H{o}s asked whether the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of maximal sum-free sets. Here, we prove the following: For each $1\leq i \leq 4$, there is a constant $C_i$ such that, given any $n\equiv i \mod 4$, $\{1, \dots , n\}$ contains $(C_i+o(1)) 2^{n/4}$ maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, a structural result of Deshouillers, Freiman, S\'os and Temkin and a recent bound on the number of subsets of integers with small sumset by Green and Morris. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.