Integer colorings with forbidden rainbow sums

Abstract

For a set of positive integers $A\subseteq [n]$, 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-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ 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.

23 pages, revised version incorporating referee comments, to appear in J. Comb. Theory Series A

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