Partitions and sums of (m,p, c)-sets

The (m, p, c)-sets characterize those sets which contain solutions to every partition-regular system of homogeneous linear equations. We show that if N is partitioned into finitely many classes there is one class and for each (m, p, c) ϵ N × N × N, an (m, p, c) set B(m, p, c) such that all finite sums choosing at most one term from each B(m, p, c) lie in the given class.