A new cardinality bound on homogeneous spaces via the Erdös-Rado theorem
by
Nathan Carlson
University of Arizona
Coauthors: Guit-Jan Ridderbos, Vrije Universiteit, The Netherlands

In 1978 E. van Douwen proved that if X is a power homogeneous Hausdorff space then |X| ≤ 2^pw(X). Given the bound 2^w(X) for general Hausdorff spaces X, this result demonstrated that well-known cardinality bounds can be improved in the presence of homogeneity. Using the Erdös-Rado Theorem, we give a substantial improvement on the van Douwen bound: if X is power homogeneous and Hausdorff, then |X| ≤ 2^c(X)pc(X). One may compare this with the cardinality bound 2^c(X)c(X), proved by Hajnal and Juhász, for an arbitrary Hausdorff space X. Our result appears to be the first application of a partition relation in the context of homogeneity.

Date received: January 9, 2008