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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.
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