Ramsey dichotomies with ordinal index
Abstract
Description
A system of uniform families on an infinite subset $M$ of $\nn$ is a collection $(\cca_ξ)_{ξ<ω_1}$ 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 $\cca_ξ$ is thin (i.e. it does not contain proper initial segments of any of its element) and the Cantor--Bendixson index, defined for $\cca_ξ$, is equal to $ξ+1$ 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).
26 pages
26 pages