On the Number of Maximal Sum-free Sets

It is shown that the set {1, 2, . . . , n} contains at most 2n/2−2n maximal sum-free subsets, provided n is large enough. A set A ⊆ [n] = {1, 2, . . . , n} is sum-free if for any two elements a, b ∈ A we have a + b / ∈ A. A sum-free set A ⊆ [n] is maximal if it is not contained in any other sum-free subset of [n]. Let s(n) and smax(n) denote the number of sum-free and maximal sum-free subsets of [n], respectively. Since the set of odd numbers is sum-free, and so is each of its subsets, s(n) ≥ 2dn/2e. It is conjectured that, in fact, we have s(n) ≤ c2 for some constant c > 0 but at this moment we know only (see Calkin [2] and Alon [1]) that the value of the exponent is close to n/2, i.e. the following holds. Theorem 1. s(n) = 2. In this note we study the behaviour of smax(n). Cameron and Erdős [4] observed that smax(n) ≥ 2bn/4c and asked if smax(n) = o(s(n)), or, maybe, even smax(n) ≤ 2n/2− , holds for some constant > 0. Our main result states that this is indeed the case. Theorem 2. There exists n0 such that for n ≥ n0 we have smax(n) ≤ 2n/2−2 n. For a maximal sum-free set A ⊆ [n] and B ⊆ A let hA(B)= ∣∣[n] \ [(A \B) ∪ ((A \B) + (A \B)) ∪ ((A \B)− (A \B)) ∪ (A \B)/2]∣∣, i.e. hA(B) denotes the number of elements i ∈ [n] one can add to A \ B so that {i} ∪ (A \ B) remains sum-free. Our argument relies on the following result of probabilistic flavor. Lemma 3. Let β be a constant such that 0 < β < 1/2. Then there exists n0 such that for n ≥ n0, every maximal sum-free set A ⊆ [n] with |A| = m ≥ n/11 contains at least n−5 √ n ( m k ) subsets B of k = bβmc elements for which hA(B) ≤ 2βn. Proof. For each i ∈ [n] \A let us choose a pair Ri = {ai, a′′ i } of two, not necessarily distinct, elements of A such that either i = ai + a ′′ i , or i = a ′ i − a′′ i , or, maybe, 2i = ai = a ′′ i (since A is a maximal sum-free set such a pair always exists). Let Received by the editors September 7, 1999 and, in revised form, December 13, 1999. 2000 Mathematics Subject Classification. Primary 11B75; Secondary 05A16. The first author was supported in part by KBN Grant 2 P03A 021 17. c ©2000 American Mathematical Society