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arXiv · 1502.07605

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

József Balogh·Hong Liu·Maryam Sharifzadeh·Andrew Treglown
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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.

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József Balogh, Hong Liu, Maryam Sharifzadeh, Andrew Treglown. 2015-02-26. Sharp bound on the number of maximal sum-free subsets of integers. https://arxiv.org/abs/1502.07605

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