Baumgartner’s isomorphism problem for ℵ2-dense suborders of R

In this paper we will analyze Baumgartner’s problem asking whether it is consistent that 2 ℵ 0 ≥ ℵ 2 and every pair of ℵ 2 -dense subsets of R are isomorphic as linear orders. The main result is the isolation of a combinatorial principle ( ∗ ∗ ) which is immune to c.c.c. forcing and which in the presence of 2 ℵ 0 ≤ ℵ 2 implies that two ℵ 2 -dense sets of reals can be forced to be isomorphic via a c.c.c. poset. Also, it will be shown that it is relatively consistent with ZFC that there exists an ℵ 2 dense suborder X of R which cannot be embedded into - X in any outer model with the same ℵ 2 .