On the Minimum Number of Monochromatic Generalized Schur Triples

The solution to the problem of finding the minimum number of monochromatic triples (x, y, x + ay) with a > 2 being a fixed positive integer over any 2-coloring of [1, n] was conjectured by Butler, Costello, and Graham (2010) and Thanathipanonda (2009). We solve this problem using a method based on Datskovsky’s proof (2003) on the minimum number of monochromatic Schur triples (x, y, x + y). We do this by exploiting the combinatorial nature of the original proof and adapting it to the general problem.