  We study universal properties of locally compact G-spaces for countable
infinite groups G. In particular we consider open invariant subsets of the
\beta-compactification of G (which is a G-space in a natural way), and their
minimal closed invariant subspaces. These are locally compact free G-spaces,
and the latter are also minimal. We examine the properies of these G-spaces
with emphasis on their universal properties.
  As an example of our resuts, we use combinatorial methods to show that each
countable infinite group admits a free minimal action on the locally compact
non-compact Cantor set.
