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- W4289104490 abstract "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 $[n]={1,ldots,n}$ 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 $B_n = { lceil frac{2}{3}(n+1) rceil, ldots, n }$. A main ingredient in the proof is a stability theorem saying that if a subset of $[n]$ of size close to $|B_n|$ 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." @default.
- W4289104490 created "2022-08-01" @default.
- W4289104490 creator A5020923873 @default.
- W4289104490 date "2018-12-22" @default.
- W4289104490 modified "2023-09-26" @default.
- W4289104490 title "Sum-free Sets of Integers with a Forbidden Sum" @default.
- W4289104490 doi "https://doi.org/10.48550/arxiv.1812.09594" @default.
- W4289104490 hasPublicationYear "2018" @default.
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