Gallai-Schur triples and related problems

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 $x + y < z$.