On the consistency of some partition theorems for continuous colorings, and the structure of aleph1-dense real order types

We present some techniques in C.C.C. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of X,-dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two Xi-dense subsets of R are isomorphic, is consistent. We e.g. prove Con(BA+ (2’0 > X,)). Let (KH, S) be the set of order types of X,-dense homogeneous subsets of R with the relation of embeddability. We prove that for every finite model (L, S) : Con(MA + (KH, C) -(L. s)) iff L is a distributive lattice. We prove that it is consistent that the Magidor-Malitz language is not countably compact. We deal with the consistency of certain topological partition theorems. E.g. We prove that MA is consistent with the axiom OCA which says: “If X is a second countable space of power K,, and {U,,, . , U,_,} is a cover of D(X)dsfX X X-{(x, x) 1 x E X} consisting of symmetric open sets, then X can be partitioned into {X, ) i E o} such that for every i E o there is 1< n such that D(Xi) E U,“. We also prove that MA + OCA j 2Ko = X,.