## Coarse and Lipschitz universality

### Volume 254 / 2021

#### Abstract

We provide several *metric universality* results. For certain classes $\mathcal C$ of metric spaces we exhibit families of metric spaces $(M_i, d_i)_{i\in I}$ which have the property that a metric space $(X,d_X)$ in $\mathcal C$ is coarsely, resp. Lipschitzly, universal for all spaces in $\mathcal C$ if $(M_i,d_i)_{i\in I}$ equi-coarsely, respectively equi-Lipschitzly, embeds into $(X,d_X)$. Such families are built as certain Schreier-type metric subsets of ${\rm c}_0$. We deduce a metric analogue of Bourgain’s theorem, which generalized Szlenk’s theorem, and prove that a space which is coarsely universal for all separable reflexive asymptotic-${\rm c} _0$ Banach spaces is coarsely universal for all separable metric spaces. One of our coarse universality results is valid under Martin’s Axiom and the negation of the Continuum Hypothesis. We discuss the strength of the universality statements that can be obtained without these additional set-theoretic assumptions. In the second part of the paper, we study universality properties of Kalton’s interlacing graphs. In particular, we prove that every finite metric space embeds almost isometrically into some interlacing graph of large enough diameter.