On the monochromatic Schur Triples type problem

TL;DR

This paper investigates the minimum number of monochromatic Schur triples in 2-colorings of [1,n], providing new bounds and proofs for specific cases, extending to multiple colors.

Contribution

It offers a new proof for the case a=1 and establishes improved upper bounds for monochromatic triples in multi-colorings.

Findings

New proof for the a=1 case of the problem.

Upper bound of n^2/(2a(a^2+2a+3)) + O(n) for a ≥ 2.

New upper bounds for r-colorings with r ≥ 3.

Abstract

We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $\{x,y,x+ay\}$ triples for $a \geq 1$. We give a new proof of the original case of $a=1$. We show that the minimum number of such triples is at most $\frac{n^2}{2a(a^2+2a+3)} + O(n)$ when $a \geq 2$. We also find a new upper bound for the minimum number, over all $r$-colorings of $[1,n]$, of monochromatic Schur triples, for $r \geq 3$.