Gallai-Schur Triples and Related Problems
Yaping Mao, Aaron Robertson, Jian Wang, Chenxu Yang, Gang Yang
TL;DR
The paper advances the study of Gallai-Schur phenomena by analyzing strict Gallai-Schur numbers, extensions of Schur-type equations with a rainbow-monochromatic paradigm, and additive inequalities. It develops both structural and counting techniques, including palindromic colorings, canonical Ramsey results, and interval-based minimization, to derive explicit thresholds (such as $n(2k)=8k+10$ and $n(2k+1)=8k+9$ for $x+y+ b=z$), asymptotic bounds on the minimum numbers of Gallai-Schur triples, and precise constants for extremal configurations in $x+y<z$. The results show that strict and ordinary Gallai-Schur numbers grow exponentially in the number of colors, with asymptotic growth rates linked to $GS(r)$ and $\,\sqrt{5}$, and provide detailed extremal constructions (including computational tools like GALRAD) that certify bounds. Together, these contributions deepen the understanding of rainbow-monochromatic trade-offs in additive Ramsey-type problems and offer concrete thresholds and asymptotics for related Diophantine colorings.
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 $x+y<z$.