Independent Deuber sets in graphs on the natural numbers

We show that for any <f>k,m,p,c,</f> if <f>G</f> is a <f>Kk</f>-free graph on <f>N</f> then there is an independent set of vertices in <f>G</f> that contains an <f>(m,p,c)</f>-set. Hence if <f>G</f> is a <f>Kk</f>-free graph on <f>N</f>, 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 <f>Ax=0</f> a solution of which can be found monochromatic under any finite coloring of <f>N</f>) and Deuber (who provided another characterization in terms of <f>(m,p,c)</f>-sets and a partition theorem for them), and of Ramsey''s theorem itself. [Copyright &y& Elsevier]