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arXiv · 2410.22024

An Unsure Note on an Un-Schur Problem

Olaf Parczyk·Christoph Spiegel
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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.

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Olaf Parczyk, Christoph Spiegel. 2024-10-29. An Unsure Note on an Un-Schur Problem. https://arxiv.org/abs/2410.22024

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