The number of maximal sum-free subsets of integers

Abstract

Cameron and Erdős [6] raised the question of how many maximal sum-free sets there are in {1, . . . , n}, giving a lower bound of 2 ⌊n/4⌋ . In this paper we prove that there are in fact at most 2 (1/4+o(1))n maximal sum-free sets in {1, . . . , n}. Our proof makes use of container and removal lemmas of Green [8,9] as well as a result of Deshouillers, Freiman, Sós and Temkin [7] on the structure of sum-free sets.A fundamental notion in combinatorial number theory is that of a sum-free set: A set S of integers is sum-free if x + y ∈ S for every x, y ∈ S (note x and y are not necessarily distinct here). The topic of sum-free sets of integers has a long history. Indeed, in 1916 Schur [19] proved that, if n is sufficiently large, then any r-colouring of [n] := {1, . . . , n} yields a monochromatic triple x, y, z such that x + y = z.Note that both the set of odd numbers in [n] and the set {⌊n/2⌋ + 1, . . . , n} are maximal sum-free sets. (A sum-free subset of [n] is maximal if it is not properly contained in another sum-free subset of [n].) By considering all possible subsets of one of these maximal sum-free sets, we see that [n] contains at least 2 ⌈n/2⌉ sum-free sets. Cameron and Erdős [5] conjectured that in fact [n] contains only O(2 n/2 ) sum-free sets. The conjecture was proven independently by Green [8] and Sapozhenko [16]. Recently, a refinement of the Cameron-Erdős conjecture was proven in [1], giving an upper bound on the number of sum-free sets in [n] of size m (for each 1 ≤ m ≤ ⌈n/2⌉).Let f (n) denote the number of sum-free subsets of [n] and f max (n) denote the number of maximal sum-free subsets of [n]. Recall that the sum-free subsets of [n] described above lie in *

The number of the maximal triangle-free graphs

“…An upper bound 2 3n/8+o(n) was proved by Wolfovitz [14]. Our proof method instantly improves this upper bound to 3 n/6+o(n) , as observed in [1]. Balogh-Liu-Sharifzadeh-Treglown [1] pushed the method further to prove a matching upper bound, 2 n/4+o(n) .…”

“…Our proof method instantly improves this upper bound to 3 n/6+o(n) , as observed in [1]. Balogh-Liu-Sharifzadeh-Treglown [1] pushed the method further to prove a matching upper bound, 2 n/4+o(n) . As [1] contains all the details, we omit further discussion here.…”

“…Balogh-Liu-Sharifzadeh-Treglown [1] pushed the method further to prove a matching upper bound, 2 n/4+o(n) . As [1] contains all the details, we omit further discussion here.…”

On solution-free sets of integers II

“…One can turn the problem of counting the number of maximal L-free subsets of [n] into one of counting maximal independent sets in an auxiliary graph. Similar techniques were used in [26,2,3,14], and in the graph setting in [5,1]. To be more precise let B and S be disjoint subsets of [n] and fix a three-variable linear equation L. The link graph L S [B] of S on B has vertex set B, and an edge set consisting of the following two types of edges:…”

On the structure of large sum-free sets of integers

“…(v) A is contained in n 5 − K, 2n 5 + K ∪ 4n 5 − K, n . Besides being interesting in their own right, these results have found several applications (see [38,26,8,9]). We remark that very few structural results are known for large sum-free sets in finite abelian groups, cf.…”