Ramsey families of subtrees of the dyadic tree

摘要

We show that for every rooted, finitely branching, pruned tree T of height. there exists a family F which consists of order isomorphic to T subtrees of the dyadic tree C = {0, 1}(< N) with the following properties: (i) the family F is a G(delta) subset of 2(C); (ii) every perfect subtree of C contains a member of F; (iii) if K is an analytic subset of F, then for every perfect subtree S of C there exists a perfect subtree S' of S such that the set {A is an element of F : A subset of S'} either is contained in or is disjoint from K.