arXiv subjects

Natasha Dobrinen

Publications and source records attributed to Natasha Dobrinen.

At least 37 records · Page 2Linked to original sources

arXiv2021

Ramsey theory of homogeneous structures: current trends and open problems

This article highlights historical achievements in the partition theory of countable homogeneous relational structures, and presents recent work, current trends, and open problems. Exciting recent developments include new methods involving logic, topological Ramsey spaces, and category theory. The paper concentrates on big Ramsey degrees, presenting their essential structure where known and outlining areas for further development. Cognate areas, including infinite dimensional Ramsey theory of homogeneous structures and partition theory of uncountable structures, are also discussed.

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arXiv2021

Fraisse classes with simply characterized big Ramsey degrees

We formulate a property strengthening the Disjoint Amalgamation Property and prove that every Fraisse structure in a finite relational language with relation symbols of arity at most two having this property has finite big Ramsey degrees which have a simple characterization. It follows that any such Fraisse structure admits a big Ramsey structure. Furthermore, we prove indivisibility for every Fraisse structure in an arbitrary finite relational language satisfying this property. This work offers a streamlined and unifying approach to Ramsey theory on some seemingly disparate classes of Fraisse structures. Novelties include a new formulation of coding trees in terms of 1-types over initial segments of the Fraisse structure, and a direct characterization of the degrees without appeal to the standard method of "envelopes".

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arXiv2021

Big Ramsey degrees of the generic partial order

As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order in a similar way as Devlin characterised big Ramsey degrees of the generic linear order (the order of rationals).

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arXiv2020

Ramsey Theory on Infinite Structures and the Method of Strong Coding Trees

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 author's 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.

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arXiv2020

The Ramsey theory of the universal homogeneous triangle-free graph Part II: Exact big Ramsey degrees

Building on previous work of the author, for each finite triangle-free graph $\mathbf{G}$, we determine the equivalence relation on the copies of $\mathbf{G}$ inside the universal homogeneous triangle-free graph, $\mathcal{H}_3$, with the smallest number of equivalence classes so that each one of the classes persists in every isomorphic subcopy of $\mathcal{H}_3$. This characterizes the exact big Ramsey degrees of $\mathcal{H}_3$. It follows that the triangle-free Henson graph is a big Ramsey structure.

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arXiv2020

Perfect Tree Forcings for Singular Cardinals

We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals $\langle κ_n: n< ω\rangle$, Prikry defined the forcing $\mathbb{P}$ all perfect subtrees of $\prod_{n<ω}κ_n$, and proved that for $κ=\sup_{n<ω}κ_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(ω,μ)$-distributive for all $μ<κ$ but $(ω,κ,δ)$-distributivity fails for all $δ<κ$, implying failure of the $(ω,κ)$-d.l. These hitherto unpublished results are included, setting the stage for the following recent results. $\mathbb{P}$ satisfies a Sacks-type property, implying that $\mathbb{B}$ is $(ω,\infty,<κ)$-distributive. The $(\mathfrak{h},2)$-d.l. and the $(\mathfrak{d},\infty,<κ)$-d.l. fail in $\mathbb{B}$. $\mathcal{P}(ω)/\mbox{Fin}$ completely embeds into $\mathbb{B}$. Also, $\mathbb{B}$ collapses $κ^ω$ to $\mathfrak{h}$. We further prove that if $κ$ is a limit of countably many measurable cardinals, then $\mathbb{B}$ adds a minimal degree of constructibility for new $ω$-sequences. Some of these results generalize to cardinals $κ$ with uncountable cofinality.

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arXiv2020

Classes of barren extensions

Henle, Mathias, and Woodin proved that, provided that $ω\rightarrow(ω)^ω$ holds in a model $M$ of ZF, then forcing with $([ω]^ω,\subseteq^*)$ over $M$ adds no new sets of ordinals, thus earning the name a "barren" extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model $M[\mathcal{U}]$, where $\mathcal{U}$ is a Ramsey ultrafilter, with many properties of the original model $M$. This begged the question of how important the Ramseyness of $\mathcal{U}$ is for these results. In this paper, we show that several classes of $σ$-closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken-Taylor ultrafilters, a class of rapid p-points of Laflamme, $k$-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares and Trujillo. Furthermore, the class of Boolean algebras $\mathcal{P}(ω^α)/\mathrm{Fin}^{\otimes α}$, $2\le α<ω_1$, forcing non-p-points also produce barren extensions.

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arXiv2019

Continuous and other finitely generated canonical cofinal maps on ultrafilters

This paper investigates conditions under which canonical cofinal maps of the following three types exist: continuous, generated by finitary end-extension preserving maps, and generated by finitary maps. The main theorems prove that every monotone cofinal map on an ultrafilter from a certain class of ultrafilters is actually canonical when restricted to some cofinal subset. These theorems are then applied to find connections between Tukey, Rudin-Keisler, and Rudin-Blass reducibilities on large classes of ultrafilters. The main theorems on canonical cofinal maps are the following. Under a mild assumption, basic Tukey reductions are inherited under Tukey reduction. In particular, every ultrafilter Tukey reducible to a p-point has continuous Tukey reductions. If $\mathcal{U}$ is a Fubini iterate of p-points, then each monotone cofinal map from $\mathcal{U}$ to some other ultrafilter is generated (on a cofinal subset of $\mathcal{U}$) by a finitary map on the base tree for $\mathcal{U}$ which is monotone and end-extension preserving - the analogue of continuous in this context. Further, every ultrafilter which is Tukey reducible to some Fubini iterate of p-points has finitely generated cofinal maps. Similar theorems also hold for some other classes of ultrafilters.

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arXiv2019

The Ramsey theory of the universal homogeneous triangle-free graph

The universal homogeneous triangle-free graph, constructed by Henson and denoted $\mathcal{H}_3$, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős-Hajnal-Posá and culminating in work of Sauer and Laflamme-Sauer-Vuksanovic, the Ramsey theory of $\mathcal{H}_3$ had only progressed to bounds for vertex colorings (Komjáth-Rödl) and edge colorings (Sauer). This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph $G$, there is a finite number $T(G)$ such that for any coloring of all copies of $G$ in $\mathcal{H}_3$ into finitely many colors, there is a subgraph of $\mathcal{H}_3$ which is again universal homogeneous triangle-free in which the coloring takes no more than $T(G)$ colors. This is the first such result for a homogeneous structure omitting copies of some non-trivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code $\mathcal{H}_3$ and the development of their Ramsey theory.

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arXiv2019

Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces

This paper investigates properties of $σ$-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 $l\ge 2$ and coloring $c:[ω]^n\rightarrow l$, there is a member $X\in\mathcal{U}$ such that the restriction of $c$ to $[X]^n$ has no more than $t$ colors. Many well-known $σ$-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 $σ$-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 $\mathcal{P}(ω^k)/\mathrm{Fin}^{\otimes k}$. 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 $σ$-closed forcings and their relationships with the classical pseudointersection number $\mathfrak{p}$.

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arXiv2019

Forcing in Ramsey theory

Ramsey theory and forcing have a symbiotic relationship. At the RIMS Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave three tutorials on Ramsey theory in forcing. The first two tutorials concentrated on forcings which contain dense subsets forming topological Ramsey spaces. These forcings motivated the development of new Ramsey theory, which then was applied to the generic ultrafilters to obtain the precise structure Rudin-Keisler and Tukey orders below such ultrafilters. The content of the first two tutorials has appeared in an expository article submitted to the SEALS 2016 Proceedings. The third tutorial concentrated on uses of forcing to prove Ramsey theorems for trees which are applied to determine big Ramsey degrees of homogeneous relational structures. This is the focus of this paper.

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arXiv2019

Forcing and the Halpern-Läuchli Theorem

We investigate the effects of various forcings on several forms of the Halpern-Läuchli Theorem. For inaccessible $κ$, we show they are preserved by forcings of size less than $κ$. Combining this with work of Zhang in \cite{Zhang17} yields that the polarized partition relations associated with finite products of the $κ$-rationals are preserved by all forcings of size less than $κ$ over models satisfying the Halpern-Läuchli Theorem at $κ$. We also show that the Halpern-Läuchli Theorem is preserved by ${{<}κ}$-closed forcings assuming $κ$ is measurable, following some observed reflection properties.

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arXiv2018

A List of Problems on the Reverse Mathematics of Ramsey Theory on the Rado Graph and on Infinite, Finitely Branching Trees

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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arXiv2018

Topological Ramsey Spaces Dense in Forcings

Topological Ramsey spaces are spaces which support infinite dimensional Ramsey theory similarly to the Ellentuck space. Each topological Ramsey space is endowed with a partial ordering which can be modified to a $σ$-closed `almost reduction' relation analogously to the partial ordering of `mod finite' on $[ω]^ω$. Such forcings add new ultrafilters satisfying weak partition relations and have complete combinatorics. In cases where a forcing turned out to be equivalent to a topological Ramsey space, the strong Ramsey-theoretic techniques have aided in a fine-tuned analysis of the Rudin-Keisler and Tukey structures associated with the forced ultrafilter and in discovering new ultrafilters with complete combinatorics.This expository paper provides an overview of this collection of results and an entry point for those interested in using topological Ramsey space techniques to gain finer insight into ultrafilters satisfying weak partition relations.

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arXiv2018

The Halpern-Läuchli Theorem at a Measurable Cardinal

Several variants of the Halpern-Läuchli Theorem for trees of uncountable height are investigated. For $κ$ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one tree on any infinite cardinal. For any finite $d \ge 2$, we prove the consistency of the Halpern-Läuchli Theorem on $d$ many $κ$-trees at a measurable cardinal $κ$, given the consistency of a $κ+d$-strong cardinal. This follows from a more general consistency result at measurable $κ$, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.

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arXiv2018

Banach Spaces from Barriers in High Dimensional Ellentuck Spaces

A new hierarchy of Banach spaces $T_k(d,θ)$, $k$ any positive integer, is constructed using barriers in high dimensional Ellentuck spaces \cite{DobrinenJSL15} following the classical framework under which a Tsirelson type norm is defined from a barrier in the Ellentuck space \cite{Argyros/TodorcevicBK}. The following structural properties of these spaces are proved. Each of these spaces contains arbitrarily large copies of $\ell_\infty^n$, with the bound constant for all $n$. For each fixed pair $d$ and $θ$, the spaces $T_k(d,θ)$, $k\ge 1$, are $\ell_p$-saturated, forming natural extensions of the $\ell_p$ space, where $p$ satisfies $dθ=d^{1/p}$. Moreover, they form a strict hierarchy over the $\ell_p$ space: For any $j<k$, the space $T_j(d,θ)$ embeds isometrically into $T_k(d,θ)$ as a subspace which is non-isomorphic to $T_k(d,θ)$.

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arXiv2016

Creature forcing and topological Ramsey spaces

This article introduces a line of investigation into connections between creature forcings and topological Ramsey spaces. Three examples of sets of pure candidates for creature forcings are shown to contain dense subsets which are actually topological Ramsey spaces. A new variant of the product tree Ramsey theorem is proved in order to obtain the pigeonhole principles for two of these examples.

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arXiv2015

Topological Ramsey spaces from Fraïssé classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlák-Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor in \cite{Baumgartner/Taylor78} generating p-points which are $k$-arrow but not $k+1$-arrow, and in a partial order of Blass in \cite{Blass73} producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of $n$ many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra $\mathcal{P}(n)$. If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space's associated ultrafilter have the structure exactly $[ω]^{<ω}$. In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template.

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