Ramsey dichotomies with ordinal index

الأصل · EN

A system of uniform families on an infinite subset M of is a collection (ξ)ξ<ω₁ of families of finite subsets of (where, ₖ consists of all k--element subset of M, for k∈) with the properties that each ξ is thin (i.e. it does not contain proper initial segments of any of its element) and the Cantor--Bendixson index, defined for 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).