Ramsey and Nash-Williams combinatorics via Schreier families

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 $\xi$.