### On Ciesielski's problems.

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Our main theorem is about iterated forcing for making the continuum larger than ℵ2. We present a generalization of [2] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ1,ℵ2 by λ, λ + (starting with λ = λ <λ > ℵ1). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ + but we can get cov(meagre) = λ and we give some applications. As in non-Cohen oracles [2], it is a “partial” countable support iteration but it is c.c.c.

We deal with Boolean algebras and their cardinal functions: π-weight π and π-character πχ. We investigate the spectrum of π-weights of subalgebras of a Boolean algebra B. Next we show that the π-character of an ultraproduct of Boolean algebras may be different from the ultraproduct of the π-characters of the factors.

For a cardinal μ we give a sufficient condition ${\oplus}_{\mu}$ (involving ranks measuring existence of independent sets) for: ${\otimes}_{\mu}$ if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| =μ) then it contains a ${2}^{{\aleph}_{0}}$-square and even a perfect square, and also for ${\otimes}_{\mu}^{\text{'}}$ if $\psi \in {L}_{{\omega}_{1},\omega}$ has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming $MA+{2}^{{\aleph}_{0}}>\mu $ for transparency, those three conditions (${\oplus}_{\mu}$, ${\otimes}_{\mu}$ and ${\otimes}_{\mu}^{\text{'}}$) are equivalent, and from this we deduce that...

We investigate σ-entangled linear orders and narrowness of Boolean algebras. We show existence of σ-entangled linear orders in many cardinals, and we build Boolean algebras with neither large chains nor large pies. We study the behavior of these notions in ultraproducts.

The main result is that for λ strong limit singular failing the continuum hypothesis (i.e. ${2}^{\lambda}>{\lambda}^{+}$), a polarized partition theorem holds.

Using a theorem from pcf theory, we show that for any singular cardinal ν, the product of the Cohen forcing notions on κ, κ < ν, adds a generic for the Cohen forcing notion on ${\nu}^{+}$.

We prove that it is consistent that the covering number of the ideal of measure zero sets has countable cofinality.

The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for ${\aleph}_{\omega}$ but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.

Let W be an inner model of ZFC. Let κ be a cardinal in V. We say that κ-covering holds between V and W iff for all X ∈ V with X ⊆ ON and V ⊨ |X| < κ, there exists Y ∈ W such that X ⊆ Y ⊆ ON and V ⊨ |Y| < κ. Strong κ-covering holds between V and W iff for every structure M ∈ V for some countable first-order language whose underlying set is some ordinal λ, and every X ∈ V with X ⊆ λ and V ⊨ |X| < κ, there is Y ∈ W such that X ⊆ Y ≺ M and V ⊨ |Y| < κ. We prove that if κ is V-regular,...

The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, $\theta ={\left({2}^{cf\left(\mu \right)}\right)}^{+}$ and ${2}^{\mu}={\mu}^{+}$ then there are Boolean algebras ${\mathbb{B}}_{1},{\mathbb{B}}_{2}$ such that $c\left({\mathbb{B}}_{1}\right)=\mu ,c\left({\mathbb{B}}_{2}\right)<\theta butc({\mathbb{B}}_{1}*{\mathbb{B}}_{2})={\mu}^{+}$. Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if $\mathbb{B}$ is a ccc Boolean algebra and ${\mu}^{{\beth}_{\omega}}\le \lambda =cf\left(\lambda \right)\le {2}^{\mu}$ then $\mathbb{B}$ satisfies the λ-Knaster condition (using the “revised GCH theorem”).

This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdotes and opinions. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references...

We sort out to a large extent when a (first order complete theory) T has a superlimit model in a cardinal λ. Also we deal with related notions of being limit.

We show the consistency of the statement: "the set of regular cardinals which are the characters of ultrafilters on ω is not convex". We also deal with the set of π-characters of ultrafilters on ω.

We prove that, e.g., if μ > cf(μ) = ℵ₀ and $\mu >{2}^{\aleph \u2080}$ and every stationary family of countable subsets of μ⁺ reflects in some subset of μ⁺ of cardinality ℵ₁, then the SCH for μ⁺ holds (moreover, for μ⁺, any scale for μ⁺ has a bad stationary set of cofinality ℵ₁). This answers a question of Foreman and Todorčević who get such a conclusion from the simultaneous reflection of four stationary sets.

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