In this paper we will analyze Baumgartner’s problem asking whether it is consistent that \ and every pair of \-dense subsets of \ 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 \ implies that two \-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 \ dense suborder X of \ which cannot be embedded into \ in any outer model with the same \. (shrink)

Measuring says that for every sequence ${\left_{\delta {\aleph _2}$. The construction works over any model of ZFC + CH and can be described as a finite support forcing iteration with systems of countable structures as side conditions and with symmetry constraints imposed on its initial segments. One interesting feature of this iteration is that it adds dominating functions $f:{\omega _1} \to {\omega _1}$ mod. countable at each stage.