An Unsure Note on an Un-Schur Problem

TL;DR

This paper explores a new anti-Ramsey variant of a classical Schur problem, establishing bounds on rainbow Schur triples in 3-colorings of integers, inspired by analogous graph theory problems.

Contribution

It introduces the first non-trivial bounds for the maximum rainbow Schur triples in 3-colorings, proposing a conjecture for the tight lower bound.

Findings

Rainbow triples can constitute at least 40% of all triples in some 3-colorings.

The maximum fraction of rainbow triples is at most approximately 66.36%.

The lower bound of 0.4 is conjectured to be tight.

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 $0.66364$. 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{o}s and S\'os regarding the maximum number of rainbow triangles in any $3$-coloring of $K_n$, which was settled by Balogh et al.