On a problem of Foreman and Magidor

A question of Foreman and Magidor asks if it is consistent for every sequence of stationary subsets of the _ns for 1n< to be mutually stationary. We get a positive answer to this question in the context of the negation of the Axiom of Choice. We also indicate how a positive answer to a generalized version of this question in a choiceless context may be obtained.The author wishes to thank James Cummings for helpful correspondence on the subject matter of this paper. The author also wishes to thank the referee and Andreas Blass, the corresponding editor, for helpful comments and suggestions that have been incorporated into this version of the paper. 03E35, 03E55 Supercompact cardinal – Indestructibility – Almost huge cardinal – Mutual stationarity – Symmetric inner modelRevised version: 6 June 2004     

Some remarks on a question of D. H. Fremlin regarding ε-density

We show the relative consistency of ℵ₁ satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω₁ satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω₁, and specifically if this and a stronger property hold under MA + ?CH. Towards this we show that if MA + ?CH holds, then for every family ? of ℵ₁ many infinite subsets of ω₁, one can find a family ? of finite subsets of ω₁ which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?. We then pose some open problems related to the question. Received: 2 June 1999 / Revised version: 2 February 2000 / Published online: 18 July 2001     

On guessing generalized clubs at the successors of regulars

König, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of a higher Souslin tree from the strong guessing principle.Complementary to the author’s work on the validity of diamond and non-saturation at the successor of singulars, we deal here with a successor of regulars. It is established that even the non-strong guessing principle entails non-saturation, and that, assuming the necessary cardinal arithmetic configuration, entails a diamond-type principle which suffices for the construction of a higher Souslin tree.We also establish the consistency of GCH with the failure of the weakest form of generalized club guessing. This, in particular, settles a question from the original paper.     

Cofinal types of ultrafilters

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 [ω₁]^<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.     

On the geometry of Urysohn's universal metric space

In recent years, much interest was devoted to the Urysohn space U and its isometry group; this paper is a contribution to this field of research. We mostly concern ourselves with the properties of isometries of U, showing for instance that any Polish metric space is isometric to the set of fixed points of some isometry φ. We conclude the paper by studying a question of Urysohn, proving that compact homogeneity is the strongest homogeneity property possible in U.     

Different versions of a first countable space without choice

We show that two versions of a first countable topological space which are equivalent in ZFC set theory split in the absence of the Axiom of Choice AC. This answers in the negative a related question from Gutierres “What is a first countable space?”.     

Small embedding characterizations for large cardinals

We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. As an application, we use such embeddings to provide new proofs of results of Christoph Weiß on the consistency strength of certain generalized tree properties. These new proofs eliminate problems contained in the original proofs provided by Weiß.     

Some remarks on a problem of J. D. Monk

In this paper we give a partial answer to a question of J. D. Monk on the size of independent families in product of interval algebras and for some related questions.The first author was supported by the United States Israel Binational Science Foundation, Publication 370The second author was supported by the Hungarian National Foundation for Scientific Research grant No. 1908     

Turing cones and set theory of the reals

We investigate Turing cones as sets of reals, and look at the relationship between Turing cones, measures, Baire category and special sets of reals, using these methods to show that Martin's proof of Turing Determinacy (every determined Turing closed set contains a Turing cone or is disjoint from one) does not work when you replace “determined” with “Blackwell determined”. This answers a question of Tony Martin. Received: 6 December 1999 / Revised version: 28 June 2000 Published online: 3 October 2001     

Countable compactness, hereditary π-character, and the Continuum Hypothesis

We prove that the Continuum Hypothesis is consistent with the statement that countably compact regular spaces that are hereditarily of countable π-character are either compact or contain an uncountable free sequence. As a corollary we solve a well-known open question by showing that the existence of a compact S-space of size greater than ℵ₁ does not follow from the Continuum Hypothesis.     

The polarized Ramsey’s theorem

We study the effective and proof-theoretic content of the polarized Ramsey’s theorem, a variant of Ramsey’s theorem obtained by relaxing the definition of homogeneous set. Our investigation yields a new characterization of Ramsey’s theorem in all exponents, and produces several combinatorial principles which, modulo bounding for formulas, lie (possibly not strictly) between Ramsey’s theorem for pairs and the stable Ramsey’s theorem for pairs. We are grateful to D. Hirschfeldt, A. Montalbán, and R. Soare for making our collaboration possible and for helpful comments and suggestions. We thank J. Schmerl for first bringing the subject of polarized partitions to our attention and J. Mileti for his generous insights. We also thank one anonymous referee for valuable observations and corrections. The first author was partially supported by an NSF Graduate Research Fellowship.     

Universality of the automorphism group of the real line

We discuss the question, whether each automorphism group (of cardinality at most ) of a linear order is embeddable into the automorphism group of the real line. We show that the answer to this question is independent of the axioms of ZFC: the answer is positive, if we assume < and Souslin's hypothesis; the answer is negative, if we assume orV=L.     

Cardinal characteristics for Menger-bounded subgroups

Machura, Shelah and Tsaban showed in [M. Machura, S. Shelah, B. Tsaban, Squares of Menger-bounded groups, Trans. Amer. Math. Soc., in press, http://arxiv.org/pdf/math.GN/0611353, 2007] that under the condition, that a relative d^′(P) of the dominating number is at least d, there are subgroups of the Baer-Specker group whose kth power is Menger-bounded and whose (k+1)st power is not. We show that the sufficient condition implies r?d and indeed can be replaced by r?d. This result includes an affirmative answer to a question by Tsaban on a possibly weaker still sufficient condition. We show that it is consistent relative to ZFC that g?r<d and there are subgroups of the Baer-Specker group whose kth power is Menger-bounded and whose (k+1)st power is not.     

Distinguishing groupwise density numbers

We show that is consistent, where is the groupwise density number and is the groupwise density number for ideals. This answers a question of Heike Mildenberger. Partially supported by Grant-in-Aid for Scientific Research (C) 17540116, Japan Society for the Promotion of Science.     

Mathias–Prikry and Laver–Prikry type forcing

We study the Mathias–Prikry and Laver–Prikry forcings associated with filters on ω. We give a combinatorial characterization of Martin?s number for these forcing notions and present a general scheme for analyzing preservation properties for them. In particular, we give a combinatorial characterization of those filters for which the Mathias–Prikry forcing does not add a dominating real.     

Scott rank of Polish metric spaces

Following the work of Friedman, Koerwien, Nies and Schlicht we positively answer their question whether the Scott rank of Polish metric spaces is countable.     

The γ-borel conjecture

In this paper we prove that it is consistent that every -set is countable while not every strong measure zero set is countable. We also show that it is consistent that every strong -set is countable while not every -set is countable. On the other hand we show that every strong measure zero set is countable iff every set with the Rothberger property is countable.Thanks to Boise State University for support during the time this paper was written and to Alan Dow for some helpful discussions and to Boaz Tsaban for some suggestions to improve an earlier version.     

Models of Cohen measurability

We show that in contrast with the Cohen version of Solovay's model, it is consistent for the continuum to be Cohen-measurable and for every function to be continuous on a non-meagre set.     

There are no hereditary productive γ-spaces

We show that if X is an uncountable productive γ-set [F. Jordan, Productive local properties of function spaces, Topology Appl. 154 (2007) 870-883], then there is a countable Y⊆X such that X?Y is not Hurewicz.Along the way we answer a question of A. Miller by showing that an increasing countable union of γ-spaces is again a γ-space. We will also show that λ-spaces with the Hurewicz property are precisely those spaces for which every co-countable set is Hurewicz.