Projective mad families

Using almost disjoint coding we prove the consistency of the existence of a definable ω-mad family of infinite subsets of ω (resp. functions from ω to ω) together with b=2^ω=ω₂.     

The cardinal characteristic for relative γ-sets

For X a separable metric space define p(X) to be the smallest cardinality of a subset Z of X which is not a relative γ-set in X, i.e., there exists an ω-cover of X with no γ-subcover of Z. We give a characterization of p(ω²) and p(ωω) in terms of definable free filters on ω which is related to the pseudo-intersection number p. We show that for every uncountable standard analytic space X that either p(X)=p(ω²) or p(X)=p(ωω). We show that the following statements are each relatively consistent with ZFC: (a) p=p(ωω)<p(ω²) and (b) p<p(ωω)=p(ω²)     

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.     

Wide scattered spaces and morasses

We show that it is relatively consistent with ZFC that ω² is arbitrarily large and every sequence s=〈sα:α<ω₂〉 of infinite cardinals with sα?ω² is the cardinal sequence of some locally compact scattered space.     

Resolvability properties via independent families

The authors give a consistent affirmative response to a question of Juhász, Soukup and Szentmiklóssy: If GCH fails, there are (many) extraresolvable, not maximally resolvable Tychonoff spaces. They show also in ZFC that for ω<λ?κ, no maximal λ-independent family of λ-partitions of κ is ω-resolvable. In topological language, that theorem translates to this: A dense, ω-resolvable subset of a space of the form (DI(λ)) is λ-resolvable.     

New basic result in classical descriptive set theory: Preservation of completeness

The aim of this note is to prove the following result:Assume that f is a continuous function from the space of irrationals ωω onto Y such that the image f(U) of every set U which is open and closed in ωω is the union of one open and one closed set. Then Y is a completely metrizable space.     

The canonical function game

The canonical function game is a game of length ω ₁ introduced by W. Hugh Woodin which falls inside a class of games known as Neeman games. Using large cardinals, we show that it is possible to force that the game is not determined. We also discuss the relationship between this result and Σ² ₂ absoluteness, cardinality spectra and Π₂ maximality for H(ω ₂) relative to the Continuum Hypothesis.     

Positional graphs and conditional structure of weakly null sequences

We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer n$n$ there are normalized weakly-null sequences of length _ωn${\omega }_n$ without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that _ωω${\omega }_{\omega }$ is the minimal cardinal κ$\kappa$ that could possibly have the property that every weakly null κ$\kappa$-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κ$\kappa$ which is smaller than the first ω$\omega$-Erd?s cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either _c0$c_0$ or _?p${?}_p$, with p≥1$p\ge 1$.     

Unconditional basic sequences in spaces of large density

We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show in particular the relative consistency of the statement that every Banach space of density ℵω contains an unconditional basic sequence.     

Cascades,order, and ultrafilters

We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. This allows us to prove (ZFC) that the class of strict _Jωω$J_{{\omega }^{\omega }}$-ultrafilters, introduced by J.E. Baumgartner in [2], is empty. We translate the result to the language of _<∞${<}_{\infty }$-sequences under an ultrafilter, investigated by C. Laflamme in [17], and we show that if there is an arbitrary long finite _<∞${<}_{\infty }$-sequence under u, then u   is at least a strict _Jωω+1$J_{{\omega }^{\omega +1}}$-ultrafilter.     

A model of Cummings and Foreman revisited

This paper concerns the model of Cummings and Foreman where from ω   supercompact cardinals they obtain the tree property at each _ℵn${\mathrm{\aleph }}_n$ for 2≤n<ω$2\le n<\omega$. We prove some structural facts about this model. We show that the combinatorics at _ℵω+1${\mathrm{\aleph }}_{\omega +1}$ in this model depend strongly on the properties of _ω1${\omega }_1$ in the ground model. From different ground models for the Cummings–Foreman iteration we can obtain either _ℵω+1∈I[_ℵω+1]${\mathrm{\aleph }}_{\omega +1}\in I[{\mathrm{\aleph }}_{\omega +1}]$ and every stationary subset of _ℵω+1${\mathrm{\aleph }}_{\omega +1}$ reflects or there are a bad scale at _ℵω${\mathrm{\aleph }}_{\omega }$ and a non-reflecting stationary subset of _ℵω+1∩cof(_ω1)${\mathrm{\aleph }}_{\omega +1}\cap \mathrm{cof}({\omega }_1)$. We also prove that regardless of the ground model a strong generalization of the tree property holds at each _ℵn${\mathrm{\aleph }}_n$ for n≥2$n\ge 2$.     

A Boolean algebra and a Banach space obtained by push-out iteration

Under the assumption that c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(ω)/fin has under CH and in the ℵ₂-Cohen model. We prove a similar result in the category of Banach spaces.     

How many miles to βX? II — Approximations to βX versus cofinal types of sets of metrics

Kada, Tomoyasu and Yoshinobu proved that the Stone-?ech compactification of a locally compact separable metrizable space is approximated by the collection of d-many Smirnov compactifications, where d is the dominating number. By refining the proof of this result, we will show that the collection of compatible metrics on a locally compact separable metrizable space has the same cofinal type, in the sense of Tukey relation, as the set of functions from ω to ω with respect to eventually dominating order.     

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.     

Linearly ordered compacta and Banach spaces with a projectional resolution of the identity

We construct a compact linearly ordered space Kω₁ of weight ℵ₁, such that the space C(Kω₁) is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, Kω₁ is a continuous image of a Valdivia compact and every separable subspace of C(Kω₁) is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.     

A measure-theoretic proof of Turing incomparability

We prove that if S is an ω-model of weak weak König’s lemma and , is incomputable, then there exists , such that A and B are Turing incomparable. This extends a recent result of Ku?era and Slaman who proved that if S₀ is a Scott set (i.e. an ω-model of weak König’s lemma) and A∈S₀, A⊆ω, is incomputable, then there exists B∈S₀, B⊆ω, such that A and B are Turing incomparable.     

Linear orderings and powers of characterizable cardinals

We say that a countable model M completely characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ⁺. If a structure M completely characterizes κ, κ is called characterizable. In this paper, we concern ourselves with cardinals that are characterizable by linearly ordered structures (cf. Definition 2.1).Under the assumption of GCH, Malitz completely resolved the problem by showing that κ is characterizable if and only if κ=ℵ_α, for some α<ω₁ (cf. Malitz (1968) [7] and Baumgartner (1974) [1]). Our results concern the case where GCH fails.From Hjorth (2002) [3], we can deduce that if κ is characterizable, then κ⁺ is characterizable by a densely ordered structure (see Theorem 2.4 and Corollary 2.5).We show that if κ is homogeneously characterizable (cf. Definition 2.2), then κ is characterizable by a densely ordered structure, while the converse fails (Theorem 2.3).The main theorems are (1) If κ>2^λ is a characterizable cardinal, λ is characterizable by a densely ordered structure and λ is the least cardinal such that κ^λ>κ, then κ^λ is also characterizable (Theorem 5.4) and (2) if ℵ_α and κ^ℵ_α are characterizable cardinals, then the same is true for κ^ℵ_α+β, for all countable β (Theorem 5.5).Combining these two theorems we get that if κ>2^ℵ_α is a characterizable cardinal, ℵ_α is characterizable by a densely ordered structure and ℵ_α is the least cardinal such that κ^ℵ_α>κ, then for all β<α+ω₁, κ^ℵ_β is characterizable (Theorem 5.7). Also if κ is a characterizable cardinal, then κ^ℵ_α is characterizable, for all countable α (Corollary 5.6). This answers a question of the author in Souldatos (submitted for publication) [8].     

On topological spaces of singular density and minimal weight

We introduce a weakening of the generalized continuum hypothesis, which we will refer to as the prevalent singular cardinals hypothesis, and show it implies that every topological space of density and weight ℵω₁ is not hereditarily Lindelöf.The assumption PSH is very weak, and in fact holds in all currently known models of ZFC.