Gallai-Schur Triples and Related Problems

Abstract: Schur's Theorem states that, for any $r \in \mathbb{Z}<sup>+$, 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 $x+y&lt;z$, and we investigate the asymptotic minimum number of rainbow and monochromatic solutions to $x+y=z$ and $x+y&lt;z$.