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Register a new userTopological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals
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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.
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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.
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.
This paper investigates properties of $sigma$-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter $mathcal{U}$ for $n$-tuples, denoted $t(mathcal{U},n)$, is the smallest number $t$ such that given any $lge 2$ and coloring $c:[omega]^nrightarrow l$, there is a member $Xinmathcal{U}$ such that the restriction of $c$ to $[X]^n$ has no more than $t$ colors. Many well-known $sigma$-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at best. In this paper, we utilize methods of topological Ramsey spaces to calculate Ramsey degrees of several classes of ultrafilters generated by $sigma$-closed forcings. These include a hierarchy of forcings due to Laflamme which generate weakly Ramsey and weaker rapid p-points, forcings of Baumgartner and Taylor and of Blass and generalizations, and the collection of non-p-points generated by the forcings $mathcal{P}(omega^k)/mathrm{Fin}^{otimes k}$. We provide a general approach to calculating the Ramsey degrees of these ultrafilters, obtaining new results as well as streamlined proofs of previously known results. In the second half of the paper, we calculate pseudointersection and tower numbers for these $sigma$-closed forcings and their relationships with the classical pseudointersection number $mathfrak{p}$.
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.
Inspired by Ramseys theorem for pairs, Rival and Sands proved what we refer to as an inside/outside Ramsey theorem: every infinite graph $G$ contains an infinite subset $H$ such that every vertex of $G$ is adjacent to precisely none, one, or infinitely many vertices of $H$. We analyze the Rival-Sands theorem from the perspective of reverse mathematics and the Weihrauch degrees. In reverse mathematics, we find that the Rival-Sands theorem is equivalent to arithmetical comprehension and hence is stronger than Ramseys theorem for pairs. We also identify a weak form of the Rival-Sands theorem that is equivalent to Ramseys theorem for pairs. We turn to the Weihrauch degrees to give a finer analysis of the Rival-Sands theorems computational strength. We find that the Rival-Sands theorem is Weihrauch equivalent to the double jump of weak K{o}nigs lemma. We believe that the Rival-Sands theorem is the first natural theorem shown to exhibit exactly this strength. Furthermore, by combining our result with a result of Brattka and Rakotoniaina, we obtain that solving one instance of the Rival-Sands theorem exactly corresponds to simultaneously solving countably many instances of Ramseys theorem for pairs. Finally, we show that the uniform computational strength of the weak Rival-Sands theorem is weaker than that of Ramseys theorem for pairs by showing that a number of well-known consequences of Ramseys theorem for pairs do not Weihrauch reduce to the weak Rival-Sands theorem. We also address an apparent gap in the literature concerning the relationship between Weihrauch degrees corresponding to the ascending/descending sequence principle and the infinite pigeonhole principle.
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