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Register a new userRamsey theory and topological dynamics for first order theories
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We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide classes of examples of theories with [pro]finite or trivial Ellis groups. As an initial motivation, we note that profiniteness of the Ellis group of a theory implies that the Kim-Pillay Galois group of this theory is also profinite, which in turn is equivalent to the equality of the Shelah and Kim-Pillay strong types. We also find several concrete examples illustrating the lack of implications between some fundamental properties. In the appendix, we give a full computation of the Ellis group of the theory of the random hypergraph with one binary and one 4-ary relation. This example shows that the assumption of NIP in the version of Newelskis conjecture for amenable theories (proved in [16]) cannot be dropped.
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In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces - called ultrahomogeneous - is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the present paper is to explore the different aspects of this connection.
We define a collection of topological Ramsey spaces consisting of equivalence relations on $omega$ with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of $omega$. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) in order to obtain a dual Ramsey theorem for such ordinals.
This article discusses some recent trends in Ramsey theory on infinite structures. Trees and their Ramsey theory have been vital to these investigations. The main ideas behind the authors recent method of trees with coding nodes are presented, showing how they can be useful both for coding structures with forbidden configurations as well as those with none. Using forcing as a tool for finite searches has allowed the development of Ramsey theory on such trees, leading to solutions for finite big Ramsey degrees of Henson graphs as well as infinite dimensional Ramsey theory of copies of the Rado graph. Possible future directions for applications of these methods are discussed.
We study the topological version of the partition calculus in the setting of countable ordinals. Let $alpha$ and $beta$ be ordinals and let $k$ be a positive integer. We write $betato_{top}(alpha,k)^2$ to mean that, for every red-blue coloring of the collection of 2-sized subsets of $beta$, there is either a red-homogeneous set homeomorphic to $alpha$ or a blue-homogeneous set of size $k$. The least such $beta$ is the topological Ramsey number $R^{top}(alpha,k)$. We prove a topological version of the ErdH{o}s-Milner theorem, namely that $R^{top}(alpha,k)$ is countable whenever $alpha$ is countable. More precisely, we prove that $R^{top}(omega^{omega^beta},k+1)leqomega^{omega^{betacdot k}}$ for all countable ordinals $beta$ and finite $k$. Our proof is modeled on a new easy proof of a weak version of the ErdH{o}s-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of $alpha$, proving among other results that $R^{top}(omega+1,k+1)=omega^k+1$, $R^{top}(alpha,k)< omega^omega$ whenever $alpha<omega^2$, $R^{top}(omega^2,k)leqomega^omega$ and $R^{top}(omega^2+1,k+2)leqomega^{omegacdot k}+1$ for all finite $k$. Our computations use a variety of techniques, including a topological pigeonhole principle for ordinals, considerations of a tree ordering based on the Cantor normal form of ordinals, and some ultrafilter arguments.
This list presents problems in the Reverse Mathematics of infinitary Ramsey theory which I find interesting but do not personally have the techniques to solve. The intent is to enlist the help of those working in Reverse Mathematics to take on such problems, and the myriad of related questions one can infer from them. A short bit of background and starting references are provided.
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