Coarse and Lipschitz universality

In this paper we provide several emph{metric universality} results. We exhibit for certain classes $cC$ of metric spaces, families of metric spaces $(M_i, d_i)_{iin I}$ which have the property that a metric space $(X,d_X)$ in $cC$ is coarsely, resp. Lipschitzly, universal for all spaces in $cC$ if the collection of spaces $(M_i,d_i)_{iin I}$ equi-coarsely, respectively equi-Lipschitzly, embeds into $(X,d_X)$. Such families are built as certain Schreier-type metric subsets of $co$. We deduce a metric analog to Bourgains theorem, which generalized Szlenks theorem, and prove that a space which is coarsely universal for all separable reflexive asymptotic-$c_0$ Banach spaces is coarsely universal for all separable metric spaces. One of our coarse universality results is valid under Martins 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 Kaltons interlacing graphs. In particular, we prove that every finite metric space embeds almost isometrically in some interlacing graph of large enough diameter.