Sum-free sets of integers
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- by H. L. Abbott and E. T. H. Wang PDF
- Proc. Amer. Math. Soc. 67 (1977), 11-16 Request permission
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 $n \leqslant 54,g(n,2) = n - [n/5]$.References
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Additional Information
- © Copyright 1977 American Mathematical Society
- 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