Schur's Theorem states that, for any r Z^+, 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.
Mao et al. (Thu,) studied this question.