On the Monochromatic Schur Triples Type Problem - Thanatipanonda, Thotsaporn "Aek" (The Electronic Journal of Combinatorics, 2009)
We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of [1,n], of monochromatic {x,y, x+ay} triples for a ≥ 1. We give a new proof of the original case of a=1. We show that the minimum number of such triples is at most n^22a(a^2+2a+3) + O(n) when a ≥ 2. We also find a new upper bound for the minimum number, over all r-colorings of [1,n], of monochromatic Schur triples, for r ≥ 3.