Some Topological Semicommutative Van Der Waerden Type Theorems and their Combinatorial Consequences
Vitaly Bergelson, Neil Hindman · Journal of the London Mathematical Society · 1992
Given k commuting actions 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.