Some Topological Semicommutative Van Der Waerden Type Theorems and their Combinatorial Consequences

Abstract

Given k commuting actions < T i 0 > o ∈ G of a group or semigroup on a compact space X we aim for the conclusion that for any open cover U we get some V ∈ D and many d ∈ G with V ∩ T 1 d V ∩ … ∩ T 1 d T 2 d … T k d V ≠ ∅ . . We obtain this result with additional hypotheses. We apply these results to obtain some van der Waerden type theorems in an arbitrary semigroup. Several counterexamples are then presented to show that both our topological and combinatorial results are the best possible.