Ramsey dichotomies with ordinal index

Abstract

A system of uniform families on an infinite subset $M$ of $\nn$ is a collection $(\cca_{\xi})_{\xi<\omega_1}$ of families of finite subsets of $\nn$ (where, $\cca_k$ consists of all $k$--element subset of $M$, for $k\in \nn$) with the properties that each $\cca_{\xi}$ is thin (i.e. it does not contain proper initial segments of any of its element) and the Cantor--Bendixson index, defined for $\cca_{\xi}$, is equal to $\xi+1$ and stable when we restrict ourselves to any subset of $M$. We indicate how to extend the generalized Schreier families to a system of uniform families. Using that notion we establish the correct (countable) ordinal index generalization of the classical Ramsey theorem (which corresponds to the finite ordinal indices).