Ramsey Theoretic Consequences of Some New Results About Algebra in the Stone-Čech Compactification

Recently [4] we have obtained some new algebraic results about βN, the Stone-Čech compactification of the discrete set of positive integers and about βW , where W is the free semigroup over a nonempty alphabet with infinitely many variables adjoined. (The results about βW extend the Graham-Rothschild Parameter Sets Theorem.) In this paper we derive some Ramsey Theoretic consequences of these results. Among these is the following, which extends the Finite Sums Theorem. Theorem. Let N be finitely colored. Then there is a color class D which is central in N and (i) there exists a pairwise disjoint collection {Di,j : i, j ∈ ω} of central subsets of D and for each i ∈ ω there exists a sequence 〈xi,n〉n=i in Di,i such that whenever F is a finite nonempty subset of ω and f : F → {1, 2, . . . ,minF} one has that Σn∈F xf(n),n ∈ Di,j where i = f(minF ) and j = f(maxF ); and (ii) at stage n when one is chosing (x0,n, x1,n, . . . , xn,n), each xi,n may be chosen as an arbitrary element of a certain central subset of Di,i, with the choice of xi,n independent of the choice of xj,n. An analogous extension of the Graham-Rothschild Theorem is established. Also included are new results about image partition regularity and kernel partition regularity of matrices.