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Article Citation - WoS: 4Citation - Scopus: 4Nonexistence of Embeddings With Uniformly Bounded Distortions of Laakso Graphs Into Diamond Graphs(Elsevier Science Bv, 2017) Ostrovska, Sofiya; Ostrovskii, Mikhail I.Diamond graphs and Laakso graphs are important examples in the theory of metric embeddings. Many results for these families of graphs are similar to each other. In this connection, it is natural to ask whether one of these families admits uniformly bilipschitz embeddings into the other. The well-known fact that Laakso graphs are uniformly doubling but diamond graphs are not, immediately implies that diamond graphs do not admit uniformly bilipschitz embeddings into Laakso graphs. The main goal of this paper is to prove that Laakso graphs do not admit uniformly bilipschitz embeddings into diamond graphs. (C) 2016 Elsevier B.V. All rights reserved.Article Citation - WoS: 2Citation - Scopus: 2Distortion of Embeddings of Binary Trees Into Diamond Graphs(Amer Mathematical Soc, 2018) Leung, Siu Lam; Nelson, Sarah; Ostrovska, Sofiya; Ostrovskii, MikhailDiamond graphs and binary trees are important examples in the theory of metric embeddings and also in the theory of metric characterizations of Banach spaces. Some results for these families of graphs are parallel to each other; for example superreflexivity of Banach spaces can be characterized both in terms of binary trees (Bourgain, 1986) and diamond graphs (Johnson-Schechtman, 2009). In this connection, it is natural to ask whether one of these families admits uniformly bilipschitz embeddings into the other. This question was answered in the negative by Ostrovskii (2014), who left it open to determine the order of growth of the distortions. The main purpose of this paper is to get a sharp up-to-a-logarithmic-factor estimate for the distortions of embeddings of binary trees into diamond graphs and, more generally, into diamond graphs of any finite branching k >= 2. Estimates for distortions of embeddings of diamonds into infinitely branching diamonds are also obtained.
