Partition regularity of \((M,P,C)\)-systems
A system of equations is partition regular over the set \(\mathbb{N}\) (positive integers) if, whenever \(\mathbb{N}\) is partitioned into finitely many classes, some one of those classes contains a solution to the given system. R. Rado characterized systems of homogeneous linear equations with integer coefficients which are parition regular. An alternate characterization was given in terms of \((m,p,c)\)-sets by \textit{W. Deuber} [Math. Z. 133, 109-123 (1973; Zbl 0254.05011)]. An \((M,P,C)\)-system is a choice of an \((m,p,c)\)-set for which \((m,p,c)\in \mathbb{N}^ 3\), together with all finite sums, choosing at most one from each \((m,p,c)\)-set. The authors show that \((M,P,C)\)-systems are partition regular. They also prove a multidimensional version of this, similar to a generalization of the Finite Sum Theorem proved by \textit{K. R. Millikan} [J. Comb. Theory, Ser. A 18, 276-290 (1975; Zbl 0323.05001)] and \textit{A. D. Taylor} [J. Comb. Theory, Ser. A 21, 137-146 (1976; Zbl 0341.05010)].
scientific article; zbMATH DE number 446852