An Unsure Note on an Un-Schur Problem
Abstract
Graham, Rödl, and Ruciński originally posed the problem of determining the minimum number of monochromatic Schur triples that must appear in any 2-coloring of the first n integers. This question was subsequently resolved independently by Datskovsky, Schoen, and Robertson and Zeilberger. Here we suggest studying a natural anti-Ramsey variant of this question and establish the first non-trivial bounds by proving that the maximum fraction of Schur triples that can be rainbow in a given 3-coloring of the first n integers is at least 0.4 and at most 0.66364. We conjecture the lower bound to be tight. This question is also motivated by a famous analogous problem in graph theory due to Erdős and Sós regarding the maximum number of rainbow triangles in any 3-coloring of Kn, which was settled by Balogh et al..