  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 \v{C}ech--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.
