$\aleph_0$-categorical Banach spaces contain $\ell_p$ or $c_0$

Abstract

This paper has three parts. First, we establish some of the basic model theoretic facts about $M_{\mathcal{T}}$, the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In~particular, we show: (1) $M_{\mathcal{T}}$ has the \emph{non independence property} (NIP); (2) every Banach space that is $\aleph_0$-categorical up to small perturbations embeds $c_0$ or $\ell_p$ ($1\leqslant p<\infty$) almost isometrically; consequently the (continuous) first-order theory of $M_{\mathcal{T}}$ does not characterize $M_{\mathcal{T}}$, up to almost isometric isomorphism.