Independent Deuber sets in graphs on the natural numbers

We show that for any k, m, p, c, if G is a Kk-free graph on N then there is an independent set of vertices in G that contains an (m, p, c)-set. Hence if G is a Kk-free graph on N, then one can solve any partition regular system of equations in an independent set. This is a common generalization of partition regularity theorems of Rado (who characterized systems of linear equations Ax = 0 a solution of which can be found monochromatic under any finite coloring of N) and Deuber (who provided another characterization in terms of (m, p, c)-sets and a partition theorem for them), and of Ramsey’s theorem itself