2025/02/28 by Mao, Yaping, Robertson, Aaron, Wang, Jian +2
#05D10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.21221
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