Given a suitable class $\mathcal K$ of finite structures, a theorem of Fraïssé shows how to construct a special countable model, called the Fraïssé limit of $\mathcal K$: the unique dense $G_\delta$ (a.k.a., "generic") isomorphism type in the space of all countable structures built from $\mathcal K$. Assuming the Continuum Hypothesis (CH), the same is true one cardinality higher: if $\mathcal K$ satisfies the hypotheses of Fraïssé's Theorem, then there is a unique generic isomorphism type in the space of all size-$\mathfrak c$ structures built from $\mathcal K$, and furthermore, these special models of size $\mathfrak c$ have properties analogous to their corresponding Fraïssé limits. Some classes of finite structures do not satisfy the hypotheses of Fraïssé's theorem, and these classes do not have Fraïssé limits. Interestingly, however, a class $\mathcal K$ may have no Fraïssé limit, but still have a unique size-$\mathfrak c$ generic model under CH. In other words, under CH, big generic models exist for any even broader range of classes than Fraïssé limits do, giving rise to Fraïssé-like uncountable structures with no true countable analogues. We will describe some aspects of the construction of these higher Fraïssé limits, and give several examples of familiar structures that can be understood in this way.
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Joint work with R. Figueroa, O. Guzmán and A. Kwela. We shall comment on Alan's construction of a countable Fréchet space of uncountable pi-weight.
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We consider the question (posed by Istvan) of whether Hereditary Lindelöf spaces of weight greater than the continuum exist in ZFC. We obtain a "provisional" solution, assuming a weakly compact cardinal, by establishing the connection to a partition relation on $[\mathfrak c]^2$. This is joint work with Istvan Juhasz.
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Topological versions of Ramsey's Theorem and the Nash-Williams Theorem suggest several new compactness properties of spaces. We will discuss recent results and open questions.
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Over 20 decades ago, Todorcevic defined a uniform filter on $\omega_1$ for each coherent Aronszajn tree $T$. He has shown that, in the presence of the Proper Forcing Axiom, this filter $\mathcal{U}(T)$ is an ultrafilter. Moreover, he showed that under these assumptions, the isomorphism type of this ultrafilter does not depend on the Aronszajn tree and that it's projection to $\omega$ is a Ramsey ultrafilter. We extend this analysis by showing that under these assumptions, $\mathcal{U}(T)$ is on one hand minimal in the Rudin-Keisler order with respect to be uniform and on the other hand, maximal with respect to Tukey's order on directed sets. This is joint work with Tom Benhamou and Luke Serafin.
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We generalize a result of Shelah that draws positive Ramsey-theoretic conclusions (the existence of infinite free sequences in algebras) from a “drastic” failure of the Singular Cardinal Hypothesis at $\aleph_\omega$. We show that the connection between these two apparently unrelated phenomena is topological and lifts to more general settings: Shelah’s theorem does not really need all of the machinery associated with pcf theory at $\aleph_\omega$. This allows us to obtain stronger results, and uncovers a dichotomy that may have further applications.
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