2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70678The 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 $({\cal A}_ξ)$ of finite subsets of the set of natural numbers, defined for every countable ordinal $ξ$.28 pages, preliminary versionFunctional AnalysisPrimary 05D10; Secondary 05C55text