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We prove that, given any ordinal $\delta < \omega_2$, there exists a transfinite $\delta$-sequence of separable Banach spaces $(X_\alpha)_{\alpha < \delta}$ such that $X_\alpha$ embeds isomorphically into $X_\beta$ and contains no subspace isomorphic to $X_\beta$ for all $\alpha < \beta < \delta$. All these spaces are subspaces of the Banach space $E_p = \bigl( \bigoplus_{n=1}^\infty \ell_p \bigr)_2$, where $1 \leq p < 2$. Moreover, assuming Martin's axiom, we prove the same for all ordinals $\delta$ of continuum cardinality. 

Abstract Main results of the paper are as follows: (1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$ . (2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs. Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique. 

Abstract Given a Banach space X and a real number α ≥ 1, we write: (1) D ( X ) ≤ α if, for any locally finite metric space A , all finite subsets of which admit bilipschitz embeddings into X with distortions ≤ C , the space A itself admits a bilipschitz embedding into X with distortion ≤ α ⋅ C ; (2) D ( X ) = α + if, for every ϵ > 0, the condition D ( X ) ≤ α + ϵ holds, while D ( X ) ≤ α does not; (3) D ( X ) ≤ α + if D ( X ) = α + or D ( X ) ≤ α. It is known that D ( X ) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) D ((⊕ n =1 ∞ X n ) p ) ≤ 1 + for every nested family of finite-dimensional Banach spaces { X n } n =1 ∞ and every 1 ≤ p ≤ ∞. (2) D ((⊕ n =1 ∞ ℓ ∞ n ) p ) = 1 + for 1 < p < ∞. (3) D ( X ) ≤ 4 + for every Banach space X with no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008). 

We prove that for every p in (1, infinity) different from 2, there exist a Banach space X isomorphic to l_p and a finite subset U in l_p, such that U is not isometric to a subset of X. This result shows that the finite isometric version of the Krivine theorem (which would be a strengthening of the Krivine theorem (1976)) does not hold.