Amalgamation and Ramsey properties of Lp spaces

We study the dynamics of the group of isometries of L-p-spaces. In particular, we study the canonical actions of these groups on the space of delta-isometric embeddings of finite dimensional subspaces of L-p(0, 1) into itself, and we show that for every real number 1 <= p < infinity with p not equal 4, 6, 8,... they are epsilon-transitive provided that delta is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fraisse Banach space which underlies these results and of which the known separable examples are the spaces L-p(0, 1), p not equal 4, 6, 8,... and the Gurarij space. We also give a proof of Ramsey property of the classes {l(p)(n)}(n), p not equal 2, infinity, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matousek and Rodl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space X with a Ramsey property of the collection of finite dimensional subspaces of X. Crown Copyright (C) 2020 Published by Elsevier Inc. All rights reserved.