Cardinal restrictions on some homogeneous compacta

We give restrictions on the cardinality of compact Hausdorff homogeneous spaces that do not use other cardinal invariants, but rather covering and separation properties. In particular, we show that it is consistent that every hereditarily normal homogeneous compactum is of cardinality $\mathfrak{c}$. We introduce property wD($\kappa$), intermediate between the properties of being weakly $\kappa$-collectionwise Hausdorff and strongly $\kappa$-collectionwise Hausdorff, and show that if $X$ is a compact Hausdorff homogeneous space in which every subspace has property wD( $\aleph _{1}$), then $X$ is countably tight and hence of cardinality $\le 2^{\mathfrak{c}}$. As a corollary, it is consistent that such a space $X$is first countable and hence of cardinality $\mathfrak{c}$. A number of related results are shown and open problems presented.