On Ramsey Numbers for Sets Free of Prescribed Differences

For a positive integer \( d \), a set \( S \) of positive integers is \emph{difference \( d \)-free} if \( |x – y| \neq d \) 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.