Abstract colorings, games and ultrafilters

The main result provide a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for natural numbers due to Milliken–Tylor, Deuber–Hindman, Bergelson–Hindman, for combinatorial covering properties due to Scheepers and Tsaban, and local properties in function spaces due to Scheepers. To this end, we use idempotent ultrafilters in the Čech–Stone compactifications of discrete infinite semigroups and topological games. The research is motivated by the recent breakthrough work of Tsaban about colorings and the Menger covering property. 1. Background 1.1. Colorings and natural numbers. A coloring of a nonempty set X is a function χ : X → {1, . . . , k}, where k is a natural number. Given a coloring χ of a set X , a set A ⊆ X is χ-monochromatic (or just monochromatic, when χ is clear from the context), if there is a color i such that χ(a) = i for all a ∈ A. By the van der Waerden Theorem [37], for each coloring of the set of natural numbers N, there is a monochromatic arithmetic progression of an arbitrarily finite length. In the comprehensive work, Bergelson and Hindman [4] considered families of finite subsets of N with the property that for each coloring of N, there is a monochromatic set in the family. By the result of Hindman [16, Theorem 6.7], a family of finite subsets of N has the above property if and only if there is an ultrafilter on N such that each set in the ultrafilter contains a set in the family. Ultrafilters play an important role in consideration of colorings of N, especially, when an algebraic structure of N is involved. Let S be an infinite semigroup with the discrete topology. Usually, we denote the semigroup operation by +, even if this operation is not commutative. The Čech–Stone compactification, βS, of S is the family of all ultrafilters on S with the topology generated by the sets { p ∈ βS : A ∈ p }, where A ⊆ S. The operation + on S can be extended to the operation + on βS such that for ultrafilters p, q ∈ βS, we have A ∈ p+ q if and only if {