  We study families of infinite block sequences of elements of the space
$\FIN_k$. In particular we study Ramsey properties of such families and Ramsey
properties localized to a selective or semiselective coideal. We show how the
stable ordered-union ultrafilters defined by Blass, and Matet-adequate families
defined by Eisworth in the case $k=1$ fit in the theory of the Ramsey space of
infinite block sequences of finite sets of natural numbers.
