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Stevo Todorcevic - One of the best experts on this subject based on the ideXlab platform.
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a new class of ramsey classification theorems and their applications in the tukey theory of Ultrafilters part 2
Transactions of the American Mathematical Society, 2013Co-Authors: Natasha Dobrinen, Stevo TodorcevicAbstract:Motivated by a Tukey classification problem we develop here a new topological Ramsey space R1 that in its complexity comes immediately after the classical Ellentuck space [8]. Associated with R1 is an ultrafilter U1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R1. This extends 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 U1: Every ultrafilter which is Tukey reducible to U1 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 U1, namely the Tukey type of a Ramsey ultrafilter.
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a new class of ramsey classification theorems and their applications in the tukey theory of Ultrafilters parts 1 and 2
Electronic Notes in Discrete Mathematics, 2013Co-Authors: Natasha Dobrinen, Stevo TodorcevicAbstract:Abstract Motivated by Tukey classification problems, we develop a new hierarchy of topological Ramsey spaces R α , α ω 1 . These spaces form a natural hierarchy of complexity, R 0 being the Ellentuck space [Erik Ellentuck, A new proof that analytic sets are Ramsey, Journal of Symbolic Logic 39 (1974), 163–165], and for each α ω 1 , R α + 1 coming immediately after R α in complexity. Associated with each R α is an ultrafilter U α , which is Ramsey for 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 R α , 1 ⩽ α ω 1 . These form a hierarchy of extensions of 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 U α , for each 1 ⩽ α ω 1 : Every nonprincipal ultrafilter which is Tukey reducible to 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 U α form a descending chain of rapid p-points of order type α + 1 .
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Cofinal types of Ultrafilters
Annals of Pure and Applied Logic, 2012Co-Authors: Dilip Raghavan, Stevo TodorcevicAbstract:Abstract We study Tukey types of Ultrafilters on ω , focusing on the question of when Tukey reducibility is equivalent to Rudin–Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many Ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated Ultrafilters includes all known Ultrafilters that are not Tukey above [ ω 1 ] ω . We give a complete characterization of all Ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given.
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Tukey types of Ultrafilters
Illinois Journal of Mathematics, 2011Co-Authors: Natasha Dobrinen, Stevo TodorcevicAbstract: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.
Natasha Dobrinen - One of the best experts on this subject based on the ideXlab platform.
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Continuous and other finitely generated canonical cofinal maps on Ultrafilters
arXiv: Logic, 2015Co-Authors: Natasha DobrinenAbstract: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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Survey on the Tukey theory of Ultrafilters
arXiv: Logic, 2014Co-Authors: Natasha DobrinenAbstract: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.
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a new class of ramsey classification theorems and their applications in the tukey theory of Ultrafilters part 2
Transactions of the American Mathematical Society, 2013Co-Authors: Natasha Dobrinen, Stevo TodorcevicAbstract:Motivated by a Tukey classification problem we develop here a new topological Ramsey space R1 that in its complexity comes immediately after the classical Ellentuck space [8]. Associated with R1 is an ultrafilter U1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R1. This extends 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 U1: Every ultrafilter which is Tukey reducible to U1 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 U1, namely the Tukey type of a Ramsey ultrafilter.
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a new class of ramsey classification theorems and their applications in the tukey theory of Ultrafilters parts 1 and 2
Electronic Notes in Discrete Mathematics, 2013Co-Authors: Natasha Dobrinen, Stevo TodorcevicAbstract:Abstract Motivated by Tukey classification problems, we develop a new hierarchy of topological Ramsey spaces R α , α ω 1 . These spaces form a natural hierarchy of complexity, R 0 being the Ellentuck space [Erik Ellentuck, A new proof that analytic sets are Ramsey, Journal of Symbolic Logic 39 (1974), 163–165], and for each α ω 1 , R α + 1 coming immediately after R α in complexity. Associated with each R α is an ultrafilter U α , which is Ramsey for 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 R α , 1 ⩽ α ω 1 . These form a hierarchy of extensions of 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 U α , for each 1 ⩽ α ω 1 : Every nonprincipal ultrafilter which is Tukey reducible to 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 U α form a descending chain of rapid p-points of order type α + 1 .
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Tukey types of Ultrafilters
Illinois Journal of Mathematics, 2011Co-Authors: Natasha Dobrinen, Stevo TodorcevicAbstract: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.
Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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cofinality spectrum problems the axiomatic approach
Topology and its Applications, 2016Co-Authors: Mary Malliaris, Saharon ShelahAbstract:Abstract Our investigations are framed by two overlapping problems: finding the right axiomatic framework for so-called cofinality spectrum problems, and a 1985 question of Dow on the conjecturally nonempty (in ZFC) region of OK but not good Ultrafilters. We define the lower-cofinality spectrum for a regular ultrafilter D on λ and show that this spectrum may consist of a strict initial segment of cardinals below λ and also that it may finitely alternate. We define so-called ‘automorphic Ultrafilters’ and prove that the Ultrafilters which are automorphic for some, equivalently every, unstable theory are precisely the good Ultrafilters. We axiomatize a bare-bones framework called “lower cofinality spectrum problems”, consisting essentially of a single tree projecting onto two linear orders. We prove existence of a lower cofinality function in this context and show by example that it holds of certain theories whose model theoretic complexity is bounded.
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model theoretic properties of Ultrafilters built by independent families of functions
Journal of Symbolic Logic, 2014Co-Authors: Mary Malliaris, Saharon ShelahAbstract:Our results in this paper increase the model-theoretic precision of a widely used method for building Ultrafilters, and so advance the general problem of constructing Ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck” in the inductive construction of a regular ultrafilter on λ (i.e., a point after which all antichains of have cardinality less than λ) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any nonlow theory. The constructions are as follows. First, we construct a regular filter on λ so that any ultrafilter extending fails to -saturate ultrapowers of the random graph, thus of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal κ, we construct a regular ultrafilter on which is λ-flexible but not -good, improving our previous answer to a question raised in Dow (1985). Third, assuming a weakly compact cardinal κ, we construct an ultrafilter to show that may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of precuts may be realized while still failing to saturate any unstable theory.
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model theoretic properties of Ultrafilters built by independent families of functions
arXiv: Logic, 2012Co-Authors: Mary Malliaris, Saharon ShelahAbstract:Our results in this paper increase the model-theoretic precision of a widely used method for building Ultrafilters, and so advance the general problem of constructing Ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, {thus} saturating any stable theory. We then prove directly that a "bottleneck" in the inductive construction of a regular ultrafilter on $\lambda$ (i.e. a point after which all antichains of $P(\lambda)/D$ have cardinality less than $\lambda$) essentially prevents any subsequent ultrafilter from being flexible, {thus} from saturating any non-low theory. The paper's three main constructions are as follows. First, we construct a regular filter $D$ on $\lambda$ so that any ultrafilter extending $D$ fails to $\lambda^+$-saturate ultrapowers of the random graph, {thus} of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal $\kappa$, we construct a regular ultrafilter on $\lambda > \kappa$ which is $\lambda$-flexible but not $\kappa^{++}$-good, improving our previous answer to a question raised in Dow 1975. Third, assuming a weakly compact cardinal $\kappa$, we construct an ultrafilter to show that $\lcf(\aleph_0)$ may be small while all symmetric cuts of cofinality $\kappa$ are realized. Thus certain families of pre-cuts may be realized while still failing to saturate any unstable theory.
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constructing regular Ultrafilters from a model theoretic point of view
arXiv: Logic, 2012Co-Authors: M Malliaris, Saharon ShelahAbstract:This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of Ultrafilters which affect saturation of unstable theories: the lower cofinality $\lcf(\aleph_0, \de)$ of $\aleph_0$ modulo $\de$, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete Ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, detected by non-low theories. Assuming $\kappa > \aleph_0$ is measurable, we construct a regular ultrafilter on $\lambda \geq 2^\kappa$ which is flexible (thus: ok) but not good, and which moreover has large $\lcf(\aleph_0)$ but does not even saturate models of the random graph. We prove that there is a loss of saturation in regular ultrapowers of unstable theories, and give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. Finally, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler's order, and by recent work of the authors on $SOP_2$. We prove that for any $n < \omega$, assuming the existence of $n$ measurable cardinals below $\lambda$, there is a regular ultrafilter $D$ on $\lambda$ such that any $D$-ultrapower of a model of linear order will have $n$ alternations of cuts, as defined below. Moreover, $D$ will $\lambda^+$-saturate all stable theories but will not $(2^{\kappa})^+$-saturate any unstable theory, where $\kappa$ is the smallest measurable cardinal used in the construction.
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The spectrum of characters of Ultrafilters on omega
arXiv: Logic, 2006Co-Authors: Saharon ShelahAbstract:We show the consistency of the set of regular cardinals which are the character of some ultrafilter on omega is not convex. We also deal with the set of pi chi-characters of Ultrafilters on omega.
Peter Krautzberger - One of the best experts on this subject based on the ideXlab platform.
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On rapid idempotent Ultrafilters
Semigroup Forum, 2014Co-Authors: Peter KrautzbergerAbstract:This short note contains the proofs of two small but somewhat surprising results about Ultrafilters on \(\mathbb {N}\): (1) strongly summable Ultrafilters are rapid, (2) every rapid ultrafilter induces a closed left ideal of rapid Ultrafilters. As a consequence, there will be rapid minimal idempotents in all models of set theory with rapid Ultrafilters. The history of this result has been published as an experiment in mathematical writing on the author’s website (Krautzberger, One Day in Colorado or Strongly Summable Ultrafilters are Rapid, 2012) and (Krautzberger, Rapid Idempotent Ultrafilters, 2012) where you can can also find additional remarks by Blass and Hindman, offering a form of peer-review.
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On rapid idempotent Ultrafilters
arXiv: Logic, 2013Co-Authors: Peter KrautzbergerAbstract:This short note contains the proofs of two small but somewhat surprising results about Ultrafilters on $\mathbb{N}$: 1. strongly summable Ultrafilters are rapid, 2. every rapid ultrafilter induces a closed left ideal of rapid Ultrafilters. As a consequence, there will be rapid minimal idempotents in all models of set theory with rapid Ultrafilters.
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On Union Ultrafilters
Order, 2011Co-Authors: Peter KrautzbergerAbstract:We present some new results on union Ultrafilters. We characterize stability for union Ultrafilters and, as the main result, we construct a new kind of unordered union ultrafilter.
Dobrinen Natasha - One of the best experts on this subject based on the ideXlab platform.
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Classes of barren extensions
'Cambridge University Press (CUP)', 2020Co-Authors: Dobrinen Natasha, Hathaway DanielAbstract:Henle, Mathias, and Woodin proved that, provided that $\omega\rightarrow(\omega)^{\omega}$ holds in a model $M$ of ZF, then forcing with $([\omega]^{\omega},\subseteq^*)$ 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 $\sigma$-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 $\mathcal{P}(\omega^{\alpha})/\mathrm{Fin}^{\otimes \alpha}$, $2\le \alpha
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Continuous and other finitely generated canonical cofinal maps on Ultrafilters
2019Co-Authors: Dobrinen NatashaAbstract: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.Comment: A few typos fixed. To appear in Fundamenta Mathematica
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Ramsey degrees of Ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces
2019Co-Authors: Dobrinen Natasha, Flores, Sonia NavarroAbstract: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 $l\ge 2$ and coloring $c:[\omega]^n\rightarrow l$, 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 $\sigma$-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 $\mathfrak{p}$.Comment: 25 page
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Topological Ramsey Spaces Dense in Forcings
2018Co-Authors: Dobrinen NatashaAbstract: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 $\sigma$-closed `almost reduction' relation analogously to the partial ordering of `mod finite' on $[\omega]^{\omega}$. 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.Comment: Expository paper submitted to the 2016 SEALS Conference Proceedings. 32 pp. Minor revision
