Gallai-Schur Triples and Related Problems

TL;DR

This paper explores Gallai-Schur numbers in various coloring problems, extending classical results to new equations and inequalities, and analyzing asymptotic behaviors of rainbow and monochromatic solutions.

Contribution

It introduces new bounds and asymptotic results for Gallai-Schur numbers in generalized equations and inequalities, expanding the monochromatic-rainbow paradigm.

Findings

Determined Gallai-Schur numbers for $x eq y$ cases.

Analyzed $x+y+b=z$ and $x+y<z$ equations.

Provided asymptotic estimates for rainbow and monochromatic solutions.

Abstract

Schur's Theorem states that, for any $r \in \mathbb{Z}^+$, there exists a minimum integer $S(r)$ such that every $r$-coloring of $\{1,2,\dots,S(r)\}$ admits a monochromatic solution to $x+y=z$. Recently, Budden determined the related Gallai-Schur numbers; that is, he determined the minimum integer $GS(r)$ such that every $r$-coloring of $\{1,2,\dots,GS(r)\}$ admits either a rainbow or monochromatic solution to $x+y=z$. In this article we consider problems that have been solved in the monochromatic setting under a monochromatic-rainbow paradigm. In particular, we investigate Gallai-Schur numbers when $x \neq y$, we consider $x+y+b=z$ and $x+y<z$, and we investigate the asymptotic minimum number of rainbow and monochromatic solutions to $x+y=z$ and $x+y<z$.