On Ramsey Numbers for Sets Free of Prescribed Differences

Abstract

For a positive integer $d$, a set $S$ of positive integers is $differenced-free$ if $|x–y|\ne d$ for all $x,y\in S$. We consider the following Ramsey-theoretical question: Given $d,k,r\in {\mathbb{Z}}^{+}$, what is the smallest integer $n$ such that every $r$-coloring of $[1,n]$ contains a monochromatic $k$-element difference $d$-free set? We provide a formula for this $n$. We then consider the more general problem where the monochromatic $k$-element set must avoid a given set of differences rather than just one difference.