Gallai-Schur Triples and Related Problems

Schur's Theorem states that, for any r is an element of & Zopf;(+), there exists a minimum integer S(r) such that every r-coloring of {1, 2, ... , 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, ... , 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 =/ 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.