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.