Infinitely many groups exhibiting intermediate growth in maximal sum-free sets

Abstract

Given an Abelian groups $G$, denote $\mu(G)$ the size of its largest sum-free subset and $f_{\max}(G)$ the number of maximal sum-free sets in $G$. Confirming a prediction by Liu and Sharifzadeh, we prove that all even-order $G\ne \mathbb{Z}_2^k$ have exponentially fewer maximal sum-free sets than $\mathbb{Z}_2^k$, i.e. $f_{\max}(G) \leq 2^{(1/2-c)\mu(G)}$, where $c > 10^{-64}$. We construct an infinite family of Abelian groups $G$ with intermediate growth in the number of maximal sum-free sets, i.e., with $ 2^{(\frac{1}{2}+c)\mu(G)}\leq f_{\max}(G) \leq 3^{(\frac{1}{3}-c)\mu(G)} $, where $c=10^{-4}$. This disproves a conjecture of Liu and Sharifzadeh and also answers a question of Hassler and Treglown in the negative. Furthermore, we determine for every even-order group $G$, the number of maximal distinct sum-free sets (where a distinct sum is $a+b= c$ with distinct $a,b,c$): it is $ 2^{(1/2+o(1))\mu(G)}$ with the only exception being $G=\mathbb{Z}_2^k \oplus \mathbb{Z}_3$, when this function is $3^{(1/3+o(1))\mu(G)}$, refuting a conjecture of Hassler and Treglown. Our proofs rely on a container theorem due to Green and Ruzsa. Another key ingredient is a sharp upper bound we establish on the number of maximal independent sets in graphs with given matching number, which interpolates between the classical results of Moon and Moser, and Hujter and Tuza. A special case of our bound implies that every $n$-vertex graph with a perfect matching has at most $2^{n/2}$ maximal independent sets, resolving another conjecture of Hassler and Treglown.