On the Monochromatic Schur Triples Type Problem

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 2 2a(a2+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.