On the Minimum Number of Monochromatic Generalized Schur Triples

TL;DR

This paper determines the minimum number of monochromatic generalized Schur triples in 2-colorings of [1,n], extending previous results on classic Schur triples using a combinatorial adaptation of Datskovsky's proof.

Contribution

It introduces a novel combinatorial method to find the minimum monochromatic triples for generalized Schur problems, generalizing prior work on the classic case.

Findings

Established the minimum number of monochromatic triples for generalized Schur problems.

Extended Datskovsky's proof technique to a broader class of triples.

Provided a new combinatorial approach for related coloring problems.

Abstract

The solution to the problem of finding the minimum number of monochromatic triples $(x,y,x+ay)$ with $a\geq 2$ being a fixed positive integer over any 2-coloring of $[1,n]$ was conjectured by Butler, Costello, and Graham (2010) and Thanathipanonda (2009). We solve this problem using a method based on Datskovsky's proof (2003) on the minimum number of monochromatic Schur triples $(x,y,x+y)$. We do this by exploiting the combinatorial nature of the original proof and adapting it to the general problem.