arXiv subjects

Natasha Dobrinen

Publications and source records attributed to Natasha Dobrinen.

47 records · Page 3Linked to original sources

arXiv2015

Infinite dimensional Ellentuck spaces and Ramsey-classification theorems

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers $B$ on $ω$ as the prototype structures, we construct a class of continuum many topological Ramsey spaces $\mathcal{E}_B$ which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projection. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces $\mathcal{E}_B$, extending the Pudlak-Rodl Theorem for barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings $\mathcal{P}([\om]^k)/\Fin^{\otimes k}$ to the countable transfinite. The $σ$-closed partial order $(\mathcal{E}_B, \sse^{\Fin^{B}})$ is forcing equivalent to $\mathcal{P}(B)/\Fin^{B}$, which forces a non-p-point ultrafilter $\mathcal{G}_B$. The present work forms the basis for further work classifying the Rudin-Keisler and Tukey structures for the hierarchy of the generic ultrafilters $\mathcal{G}_B$.

math.LO↗

arXiv2015

Spectra of Tukey types of ultrafilters on Boolean algebras

Extending recent investigations on the structure of Tukey types of ultrafilters on $\mathcal{P}(ω)$ to Boolean algebras in general, we classify the spectra of Tukey types of ultrafilters for several classes of Boolean algebras, including interval algebras, tree algebras, and pseudo-tree algebras.

math.LO↗

arXiv2014

Rainbow Ramsey simple structures

A relational structure $\mathrm{R}$ is {\em rainbow Ramsey} if for every finite induced substructure $\mathrm{C}$ of $\mathrm{R}$ and every colouring of the copies of $\mathrm{C}$ with countably many colours, such that each colour is used at most $k$ times for a fixed $k$, there exists a copy $\mathrm{R}^\ast$ of $\mathrm{R}$ so that the copies of $\mathrm{C}$ in $\mathrm{R^\ast}$ use each colour at most once. We show that certain ultrahomogenous binary relational structures, for example the Rado graph, are rainbow Ramsey. Via compactness this then implies that for all finite graphs $\mathrm{B}$ and $\mathrm{C}$ and $k \in ω$, there exists a graph $\mathrm{A}$ so that for every colouring of the copies of $\mathrm{C}$ in $\mathrm{A}$ such that each colour is used at most $k$ times, there exists a copy $\mathrm{B}^\ast$ of $\mathrm{B}$ in $\mathrm{A}$ so that the copies of $\mathrm{C}$ in $\mathrm{B^\ast}$ use each colour at most once.

math.CO↗

arXiv2014

Topological Ramsey spaces and metrically Baire sets

We characterize a class of topological Ramsey spaces such that each element $\mathcal R$ of the class induces a collection $\{\mathcal R_k\}_{k<ω}$ of projected spaces which have the property that every Baire set is Ramsey. Every projected space $\mathcal R_k$ is a subspace of the corresponding space of length-$k$ approximation sequences with the Tychonoff, equivalently metric, topology. This answers a question of S. Todorcevic and generalizes the results of Carlson \cite{Carlson}, Carlson-Simpson \cite{CarSim2}, Prömel-Voigt \cite{PromVoi}, and Voigt \cite{Voigt}. We also present a new family of topological Ramsey spaces contained in the aforementioned class which generalize the spaces of ascending parameter words of Carlson-Simpson \cite{CarSim2} and Prömel-Voigt \cite{PromVoi} and the spaces $\FIN_m^{[\infty]}$, $0<m<ω$, of block sequences defined by Todorcevic \cite{Todo}.

math.CO↗

arXiv2014

High dimensional Ellentuck spaces and initial chains in the Tukey structure of non-p-points

The generic ultrafilter $\mathcal{G}_2$ forced by $\mathcal{P}(ω\timesω)/($Fin$\otimes$Fin) was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters (in a recent paper of Blass, Dobrinen, and Raghavan), but it was left open where exactly in the Tukey order it lies. We prove that $\mathcal{G}_2$ is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each $k\ge 2$, the collection of all nonprincipal ultrafilters Tukey reducible to the generic ultrafilter $\mathcal{G}_k$ forced by $\mathcal{P}(ω^k)/$Fin$^{\otimes k}$ forms a chain of length $k$. Essential to the proof is the extraction of a dense subset $\mathcal{E}_k$ from (Fin$^{\otimes k})^+$ which we prove to be a topological Ramsey space. The spaces $\mathcal{E}_k$, $k\ge 2$, form a hiearchy of high dimensional Ellentuck spaces. New Ramsey-classification theorems for equivalence relations on fronts on $\mathcal{E}_k$ are proved, extending the Pudlak-Rodl Theorem for fronts on the Ellentuck space, which are applied to find the Tukey structure below $\mathcal{G}_k$.

math.LO↗

arXiv2014

Survey on the Tukey theory of ultrafilters

This article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell's Problem and the question of how closely related the Rudin-Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guarantee that an ultrafilter is below the Tukey maximum. The known canonical forms for cofinal maps on ultrafilters are reviewed, as well as their applications to finding which structures embed into the Tukey types of ultrafilters. With the addition of some Ramsey theory, fine analyses of the structures at the bottom of the Tukey hierarchy are made.

math.LO↗

arXiv2013

The next best thing to a P-point

We study ultrafilters on $ω^2$ produced by forcing with the quotient of $\scr P(ω^2)$ by the Fubini square of the Fréchet filter on $ω$. We show that such an ultrafilter is a weak P-point but not a P-point and that the only non-principal ultrafilters strictly below it in the Rudin-Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above $[ω_1]^{<ω}$, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of non-isomorphic selective ultrafilters.

math.LO↗

arXiv2012

A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters

Motivated by Tukey classification problems and building on work in \cite{Dobrinen/Todorcevic11}, we develop a new hierarchy of topological Ramsey spaces $\mathcal{R}_α$, $α<ω_1$. These spaces form a natural hierarchy of complexity, $\mathcal{R}_0$ being the Ellentuck space, and for each $α<ω_1$, $\mathcal{R}_{α+1}$ coming immediately after $\mathcal{R}_α$ in complexity. Associated with each $\mathcal{R}_α$ is an ultrafilter $\mathcal{U}_α$, which is Ramsey for $\mathcal{R}_α$, and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on $\mathcal{R}_α$, $2\leα<ω_1$. These are analogous to the Pudlak-\Rodl\ Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our Ramsey-classification theorems to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to $\mathcal{U}_α$, for each $2\leα<ω_1$: Every ultrafilter which is Tukey reducible to $\mathcal{U}_α$ is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of rapid p-points. Moreover, we show that the Tukey types of nonprincipal ultrafilters Tukey reducible to $\mathcal{U}_α$ form a descending chain of order type $α+1$.

math.LO↗

arXiv2012

A Ramsey-Classification Theorem and its Application in the Tukey Theory of Ultrafilters

Motivated by a Tukey classification problem we develop here a new topological Ramsey space $\mathcal{R}_1$ that in its complexity comes immediately after the classical is a natural Ellentuck space \cite{MR0349393}. Associated with $\mathcal{R}_1$ is an ultrafilter $\mathcal{U}_1$ which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on $\mathcal{R}_1$. This is analogous to the Pudlak-\Rodl\ Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our canonization theorem to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to $\mathcal{U}_1$: Every ultrafilter which is Tukey reducible to $\mathcal{U}_1$ is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of ultrafilters. Moreover, we show that there is exactly one Tukey type of nonprincipal ultrafilters strictly below that of $\mathcal{U}_1$, namely the Tukey type a Ramsey ultrafilter.

math.LO↗

arXiv2011

Tukey types of ultrafilters

We investigate the structure of the Tukey types of ultrafilters on countable sets partially ordered by reverse inclusion. A canonization of cofinal maps from a p-point into another ultrafilter is obtained. This is used in particular to study the Tukey types of p-points and selective ultrafilters. Results fall into three main categories: comparison to a basis element for selective ultrafilters, embeddings of chains and antichains into the Tukey types, and Tukey types generated by block-basic ultrafilters on FIN.

math.LO↗

arXiv2011

Continuous cofinal maps on ultrafilters

An ultrafilter $\mathcal{U}$ on a countable base {\em has continuous Tukey reductions} if whenever an ultrafilter $\mathcal{V}$ is Tukey reducible to $\mathcal{U}$, then every monotone cofinal map $f:\mathcal{U}\ra\mathcal{V}$ is continuous when restricted to some cofinal subset of $\mathcal{U}$. In the first part of the paper, we give mild conditions under which the property of having continuous Tukey reductions is inherited under Tukey reducibility. In particular, if $\mathcal{U}$ is Tukey reducible to a p-point then $\mathcal{U}$ has continuous Tukey reductions. In the second part, we show that any countable iteration of Fubini products of p-points has Tukey reductions which are continuous with respect to its topological Ramsey space of $\vec{\mathcal{U}}$-trees.

math.LO↗