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## Coarse and Lipschitz universality

Fundamenta Mathematicae MSC: Primary 46B06, 46B20, 46B85, 46T99, 05C63. DOI: 10.4064/fm956-9-2020 Opublikowany online: 11 December 2020

#### Streszczenie

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.

#### Autorzy

• Florent BaudierDepartment of Mathematics
Texas A&M University
College Station, TX 77843, U.S.A.
e-mail
• Gilles LancienLaboratoire de Mathématiques de Besançon
Université Bourgogne Franche-Comté
16 route de Gray
25030 Besançon Cédex, France
e-mail
• Pavlos MotakisDepartment of Mathematics and Statistics
York University
4700 Keele Street
e-mail
• Thomas SchlumprechtDepartment of Mathematics
Texas A&M University
College Station
TX 77843-3368, U.S.A.
and
Faculty of Electrical Engineering
Czech Technical University in Prague
Zikova 4
166 27 Praha, Czech Republic
e-mail

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