TY - JOUR

T1 - A sharp bound on the number of maximal sum-free sets

AU - Balogh, József

AU - Liu, Hong

AU - Sharifzadeh, Maryam

AU - Treglown, Andrew

N1 - Funding Information:
1 Email: jobal@math.uiuc.edu. Research is partially supported by Simons Fellowship, NSA Grant H98230-15-1-0002, NSF CAREER Grant DMS-0745185, Arnold O. Beckman Research Award (UIUC Campus Research Board 15006) and Marie Curie FP7-PEOPLE-2012-IIF 327763 2 Email: hliu36@illinois.edu. 3 Email: sharifz2@illinois.edu. 4 Email: a.c.treglown@bham.ac.uk.
Publisher Copyright:
© 2015 Elsevier B.V.

PY - 2015/11

Y1 - 2015/11

N2 - Cameron and Erdős 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. We prove the following: For each 1≤i≤4, there is a constant Ci such that, given any n≡i mod 4, {1,..., n} contains (Ci+o(1))2n/4 maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, a structural result of Deshouillers, Freiman, Sós and Temkin and a recent bound on the number of subsets of integers with small sumset by Green and Morris.

AB - Cameron and Erdős 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. We prove the following: For each 1≤i≤4, there is a constant Ci such that, given any n≡i mod 4, {1,..., n} contains (Ci+o(1))2n/4 maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, a structural result of Deshouillers, Freiman, Sós and Temkin and a recent bound on the number of subsets of integers with small sumset by Green and Morris.

KW - Container method

KW - Independent sets

KW - Sum-free sets

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U2 - 10.1016/j.endm.2015.06.010

DO - 10.1016/j.endm.2015.06.010

M3 - Article

AN - SCOPUS:84947807826

VL - 49

SP - 57

EP - 64

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

ER -