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An Unsure Note on an Un-Schur Problem

Olaf Parczyk, Christoph Spiegel

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 $0.66656$. 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 $K_n$, which was settled by Balogh et al.

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 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 -coloring of the first integers is at least and at most . 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 -coloring of , which was settled by Balogh et al.

Paper Structure

This paper contains 7 sections, 3 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of \ref{['thm:rainbow-schur-triple']}
  3. The lower bound
  4. The upper bound
  5. Concluding remarks
  6. Lower bound
  7. Upper bound

Key Result

Theorem 1.1

The maximum fraction of Schur triples that can be rainbow in a $3$-coloring of the first $n$ integers asymptotically is between $0.4$ and $0.66656$.

Figures (1)

  • Figure 1: Region of feasible $\alpha$ and $\beta$ for fixed $\gamma_0= 0.077102$ with the first bound from \ref{['eq:second_bound']} in blue and the second bound from \ref{['eq:third_bound']} in green. The optimal point is in red with the corresponding contour line for the objective from \ref{['eq:SRupper']}. The black dot corresponds to our lower bound construction.

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof