Geodesic
Sharp bound on the number of maximal sum-free subsets of integers
Journal of the European Mathematical Society, Tome 20 (2018) no. 8, pp. 1885-1911

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Cameron and Erdős [6] asked whether the number of maximal sum-free sets in {1,...,n} is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2⌊n/4⌋ for the number of maximal sum-free sets. Here, we prove the following: For each 1≤i≤4, there is a constant Ci​ such that, given any n≡imod4, {1,...,n} contains (Ci​+o(1))2n/4 maximal sum-free sets. Our proof makes use of container and removal lemmas of Green [11, 12], a structural result of Deshouillers, Freiman, Sós and Temkin [7] and a recent bound on the number of subsets of integers with small sumset by Green and Morris [13]. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.
DOI : 10.4171/jems/802
Classification : 11-XX, 05-XX
Keywords: Sum-free sets, Independent sets, container method 
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     title = {Sharp bound on the number of maximal sum-free subsets of integers},
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József Balogh; Hong Liu; Maryam Sharifzadeh; Andrew Treglown. Sharp bound on the number of maximal sum-free subsets of integers. Journal of the European Mathematical Society, Tome 20 (2018) no. 8, pp. 1885-1911. doi: 10.4171/jems/802

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