On the consistency of some partition theorems for continuous colorings, and the structure of א₁-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 א₁-dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two א₁-dense subsets of R are isomorphic, is consistent. We e.g. prove Con(BA+(2^א₀>א₂)). Let <K^H,<> be the set of order types of א₁-dense homogeneous subsets of R with the relation of embeddability. We prove that for every finite model <L, <->: Con(MA+ <K^H, <-> {difference between} <L, <->) 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 א₁, and {U₀,\h.;,U_n-1} is a cover of D(X){A figure is presented}XxX-}<x,x>|xε{lunate}X} consisting of symmetric open sets, then X can be partitioned into {X_i \brvbar; i ε{lunate} ω} such that for every i ε{lunate} ω there is l<n such that D(X_i)⊇U_l". We also prove that MA+OCA [xrArr] 2 א₀ = א₂.