Homogeneous countably compact spaces X and Y whose product X×Y is
not pseudocompact are constructed. It is proved that all compact subsets of
homogeneous subspaces of the third power of an extremally disconnected space
are finite. Moreover, under CH, all compact subsets of homogeneous subspaces of
any finite power of an extremally disconnected space are finite and all compact
subsets of homogeneous subspaces of the countable power of an extremally
disconnected space are metrizable. It is also proved that all compact
homogeneous subspaces of finite powers of an extremally disconnected space are
finite, which strengthens Frol\'{\i}k's theorem