Recent Developments in Finite Ramsey Theory: Foundational Aspects and Connections with Dynamics

We survey some recent results in Ramsey theory. We indicate their connections with topological dynamics. On the foundational side, we describe an abstract approach to finite Ramsey theory. We give one new application of the abstract approach through which we make a connection with the theme of duality in Ramsey theory. We finish with some open problems. 1. Ramsey theory and topological dynamics Recent years have seen a renewed interest in Ramsey theory that lead to advances both in proving new concrete Ramsey results and in developing the foundational aspects of the theory. To a large extent this interest in Ramsey theory was sparked by the discovery of its close connections with topological dynamics and especially with the notion of extreme amenability and related to it problem of computing universal minimal flows of topological groups. A topological group is called extremely amenable if each continuous action of it on a compact (always assumed Hausdorff) space has a fixed point. First such groups were discovered by Herer and Christensen [13] using functional analytic methods. It was then shown by Veech [40] that extremely amenable groups cannot be locally compact. It turned out, however, that some very interesting groups are extremely amenable; for example, Gromov and Milman [12] showed that the unitary group of a separable infinite dimensional Hilbert space, taken with the strong operator topology and with composition as the group operation, is extremely amenable. The proof in [12] of this theorem used probabilistic methods of concentration of measure through the notion of Lévy group. (Lévy groups are topological groups possessing an increasing sequence of compact subgroups with dense union and with concentration of measure exhibited by the sequence of the normalized Haar measures on the compact subgroups.) Concentration of measure grew to be one of the two main methods used in proving extreme amenability. It was not until Pestov’s paper [27] that the second general method—Ramsey theory—was discovered. Pestov showed that the group of all increasing bijections from Q to itself, with pointwise convergence topology and composition as the group operation, is extremely amenable. His proof used the classical Ramsey theorem in a way that appeared, as it turned out correctly, fundamental. Pestov’s article was Research supported by NSF grant DMS-1266189.