SUM-FREE SETS OF INTEGERS WITH A FORBIDDEN SUM

A set of integers is sum-free if it contains no solution to the equation x + y = z. We study sum-free subsets of the set of integers [n] = {1,...n} for which the integer 2n + 1 cannot be represented as a sum of their elements. We prove a bound of O(2(n/3)) on the number of these sets, which matches, up to a multiplicative constant, the lower bound obtained by considering all subsets of B-n = {inverted right perperndicular2/3 (n + 1)inverted left perpendicular,..., n}. A main ingredient in the proof is a stability theorem saying that if a subset of [n] of size close to vertical bar B-n vertical bar contains only a few subsets that contradict the sum-freeness or the forbidden sum, then it is almost contained in B-n. Our results are motivated by the question of counting symmetric complete sum-free subsets of cyclic groups of prime order. The proofs involve Freiman's 3k - 4 theorem, Green's arithmetic removal lemma, and structural results on independent sets in hypergraphs.