Integer colorings with forbidden rainbow sums

Abstract

For a set of positive integers $A\subseteq \left[n\right]$, an r-coloring of A is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothschild problem in the context of sum-free sets, which asks for the subsets of $\left[n\right]$ with the maximum number of rainbow sum-free r-colorings. We show that for $r=3$, the interval $\left[n\right]$ is optimal, while for $r\ge 8$, the set $[\lfloor n/2\rfloor ,n]$ is optimal. We also prove a stability theorem for $r\ge 4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.