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Abstract colorings, games and ultrafilters (2107.02830v1)
Published 6 Jul 2021 in math.CO and math.GN
Abstract: 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.