An unsure talk on an un-Schur problem

Abstract: Graham, R\" odl, and Ruci\' nski 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 0.66656. We conjecture the lower bound to be tight. This question is also motivated by a famous analogous problem in graph theory due to Erd\H os and S\' os regarding the maximum number of rainbow triangles in any 3-coloring of $K_n$, which was settled by Balogh, et al. This is joint work with Olaf Parczyk.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)