Test-space characterizations of some classes of Banach spaces

Let $\mathcal{P}$ be a class of Banach spaces and let $T=\{T_\alpha\}_{\alpha\in A}$ be a set of metric spaces. We say that $T$ is a set of {\it test-spaces} for $\mathcal{P}$ if the following two conditions are equivalent: (1) $X\notin\mathcal{P}$; (2) The spaces $\{T_\alpha\}_{\alpha\in A}$ admit uniformly bilipschitz embeddings into $X$. The first part of the paper is devoted to a simplification of the proof of the following test-space characterization obtained in M.I. Ostrovskii [Different forms of metric characterizations of classes of Banach spaces, Houston J. Math., to appear]: For each sequence $\{X_m\}_{m=1}^\infty$ of finite-dimensional Banach spaces there is a sequence $\{H_n\}_{n=1}^\infty$ of finite connected unweighted graphs with maximum degree 3 such that the following conditions on a Banach space $Y$ are equivalent: (A) $Y$ admits uniformly isomorphic embeddings of $\{X_m\}_{m=1}^\infty$; (B) $Y$ admits uniformly bilipschitz embeddings of $\{H_n\}_{n=1}^\infty$. The second part of the paper is devoted to the case when $\{X_m\}_{m=1}^\infty$ is an increasing sequence of spaces. It is shown that in this case the class of spaces given by (A) can be characterized using one test-space, which can be chosen to be an infinite graph with maximum degree 3.