  The main results of this paper (a) extend the finite Ramsey partition
theorem, and (b) employ this extension to obtain a stronger form of the
infinite Nash-Williams partition theorem, and also a new proof of Ellentuck's,
and hence Galvin-Prikry's partition theorem. The proper tool for this
unification of the classical partition theorems at a more general and stronger
level is the system of Schreier families $({\cal A}_{\xi})$ of finite subsets
of the set of natural numbers, defined for every countable ordinal $\xi$.
