The number of maximal sum-free subsets of integers

Balogh, József; Liu, Hong; Sharifzadeh, Maryam; Treglown, Andrew

doi:10.1090/S0002-9939-2015-12615-9

The number of maximal sum-free subsets of integers
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by József Balogh, Hong Liu, Maryam Sharifzadeh and Andrew Treglown

Proc. Amer. Math. Soc. 143 (2015), 4713-4721

DOI: https://doi.org/10.1090/S0002-9939-2015-12615-9

Published electronically: April 2, 2015

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Abstract:

Cameron and Erdős 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, Sós and Temkin on the structure of sum-free sets.

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