Sum-free sets of integers

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 $n \leqslant 54,g(n,2) = n - [n/5]$.