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Journal of the London Mathematical Society 1992 s2-45(3):385-403; doi:10.1112/jlms/s2-45.3.385
© 1992 by London Mathematical Society
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© Oxford University Press

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

Vitaly Bergelson and Neil Hindman

Department of Mathematics, Ohio State University 231 West 18th Avenue, Columbus, Ohio 43210-1174, USA
Department of Mathematics, Howard University Washington, DC 20059, USA

Given k commuting actions Formula 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 isin D and many d isin G with Formula. 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.


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