Abstract
Cameron and Erdos [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 2n/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 ≡ 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 [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.
Original language | English (US) |
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Pages (from-to) | 1885-1911 |
Number of pages | 27 |
Journal | Journal of the European Mathematical Society |
Volume | 20 |
Issue number | 8 |
DOIs | |
State | Published - 2018 |
Keywords
- Container method
- Independent sets
- Sum-free sets
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics