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Sum-free sets of integers


Authors: H. L. Abbott and E. T. H. Wang
Journal: Proc. Amer. Math. Soc. 67 (1977), 11-16
MSC: Primary 10L05; Secondary 05A17
DOI: https://doi.org/10.1090/S0002-9939-1977-0485759-7
MathSciNet review: 0485759
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Abstract: A set S of integers is said to be sum-free if $ a,b \in S$ implies $ a + b \notin S$. In this paper, we investigate two new problems on sum-free sets: (1) Let $ f(k)$ denote the largest positive integer for which there exists a partition of $ \{ 1,2, \ldots ,f(k)\} $ into k sum-free sets, and let $ h(k)$ denote the largest positive integer for which there exists a partition of $ \{ 1,2, \ldots ,h(k)\} $ into k sets which are sum-free $ \bmod h(k) + 1$. We obtain evidence to support the conjecture that $ f(k) = h(k)$ for all k. (2) Let $ g(n,k)$ denote the cardinality of a largest subset of $ \{ 1,2, \ldots ,n\} $ that can be partitioned into k sum-free sets. We obtain upper and lower bounds for $ g(n,k)$. We also show that $ g(n,1) = [(n + 1)/2]$ and indicate how one may show that for all $ n \leqslant 54,g(n,2) = n - [n/5]$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0485759-7
Keywords: Sum-free set, k-wise sum-free set
Article copyright: © Copyright 1977 American Mathematical Society

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