Metric Baumgartner theorems and universality (Q2459310)

The paper was motivated by a classical result of Baumgartner: It is consistent that \(2^{\aleph_0}=\aleph_{2}\) and any two \(\aleph_{1}\)-dense subsets of \(\mathbb R\) are order isomorphic to each other. Another result related was obtained by M. Kojman and S. Shelah who proved that if \(X\) is a metric space which is almost isometric to the Urysohn space \(\mathbb{U}\) then they are in fact isometric. Two metric spaces \(X\) and \(Y\) are almost isometric if for every \(\lambda>1\) there is a homeomorphism \(f:X\rightarrow Y\) such that \(f\) and \(f^{-1}\) are Lipschitz maps with constant \(\lambda.\) The Urysohn space \(\mathbb{U}\) is the unique (up to isometry) separable metric space which is universal and ultrahomogeneous (meaning that \(\mathbb{U}\) contains an isometric copy of every separable metric space and every isometry between two finite metric subspaces of \(\mathbb{U}\) can be extended to an isometry of \(\mathbb{U}\) onto itself). In the paper under review the authors show that if \textsl{ZFC} is consistent, then so is \textsl{ZFC} + \(\mathfrak{c}=\aleph_{2}\) + ``for every metric space \(X\), if \(X\) is isometric to the Urysohn space or to some separable Banach space then \(X\) has a nowhere meager subset of size \(\aleph_{1}\) and any two nowhere meager subsets of \(X\) of size \(\aleph_{1}\) are almost isometric to each other''. A very nice consequence is the consistency of an almost-isometry universal separable metric space of cardinality \(\aleph_{1}<\mathfrak{c}\). The proof of their main result uses Shelah's oracle forcing method first adding with finite conditions sufficiently many bi-Lipschitz bijections between nowhere meager subsets of size \(\aleph_{1}\) of a separable Banach space. The authors end their paper stating the central open problem: Is it consistent to have an almost-isometry universal metric space in cardinality \(\aleph_{1}<2^{\aleph_0}\)? They also state: Is it consistent that there is a nowhere meager subset of \(\mathbb{U}\) of cardinality \(\aleph_{2}\) and that any two nowhere meager subsets of \(\mathbb{U}\) of cardinality \(\aleph_{2}\) are almost isomorphic?