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Coarse and Lipschitz universality

Volume 254 / 2021

Florent Baudier, Gilles Lancien, Pavlos Motakis, Thomas Schlumprecht Fundamenta Mathematicae 254 (2021), 181-214 MSC: Primary 46B06, 46B20, 46B85, 46T99, 05C63. DOI: 10.4064/fm956-9-2020 Published online: 11 December 2020


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.


  • Florent BaudierDepartment of Mathematics
    Texas A&M University
    College Station, TX 77843, U.S.A.
  • Gilles LancienLaboratoire de Mathématiques de Besançon
    Université Bourgogne Franche-Comté
    16 route de Gray
    25030 Besançon Cédex, France
  • Pavlos MotakisDepartment of Mathematics and Statistics
    York University
    4700 Keele Street
    Toronto, Ontario M3J 1P3, Canada
  • Thomas SchlumprechtDepartment of Mathematics
    Texas A&M University
    College Station
    TX 77843-3368, U.S.A.
    Faculty of Electrical Engineering
    Czech Technical University in Prague
    Zikova 4
    166 27 Praha, Czech Republic

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