Related papers: Indestructibility of the tree property
Let $\mathbb{B}$ be a complete Boolean algebra. We show, as an application of a previous result of the author, that if $\lambda$ is an infinite cardinal and $\mathbb{B}$ is weakly $(\lambda^\omega, \omega)$-distributive, then $\mathbb{B}$…
Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $\lambda>\kappa$ to $\kappa$) such that the composition still…
We investigate the generalized tree properties and guessing model properties introduced by Wei\ss\ and Viale, as well as natural weakenings thereof, studying the relationships among these properties and between these properties and other…
We prove that every point-finite family of nonempty functionally open sets in a topological space $X$ has the cardinality at most an infinite cardinal $\kappa$ if and only if $w(X)\leq\kappa$ for every Valdivia compact space $Y\subseteq…
We show that Vopenka's Principle and Vopenka cardinals are indestructible under reverse Easton forcing iterations of increasingly directed-closed partial orders, without the need for any preparatory forcing. As a consequence, we are able to…
In this paper, we demonstrate that if, for every $\kappa$-complete fine filter $F$ over $\mathcal{P}_{\kappa}\lambda$, the associated Namba forcing $\mathrm{Nm}(\kappa,\lambda,F)$ is semiproper, then $\square(\mu,{<}\aleph_1)$ fails for all…
Motivated by two open questions about two-cardinal tree properties, we introduce and study generalized narrow system properties. The first of these questions asks whether the strong tree property at a regular cardinal $\kappa \geq \omega_2$…
A cardinal lambda is called omega-inaccessible if for all mu < lambda we have mu^omega<lambda. We show that for every omega-inaccessible cardinal lambda there is a CCC (hence cardinality and cofinality preserving) forcing that adds a…
We continue the development of the theory of construction schemes over $\omega_1$ as introduced by the third author by studying their relation with forcing axioms. Formally, we introduce the cardinals $\mathfrak{m}^n_{\mathcal{F}}$ and use…
We will consider a number of new large-cardinal properties, the $\alpha$-tremendous cardinals for each limit ordinal $\alpha>0$, the hyper-tremendous cardinals, the $\alpha$-enormous cardinals for each limit ordinal $\alpha>0$, and the…
Suppose that lambda = mu^+. We consider two aspects of the square property on subsets of lambda. First, we have results which show e.g. that for aleph_0 <= kappa =cf (kappa)< mu, the equality cf([mu]^{<= kappa}, subseteq)= mu is a…
Foreman proved the Duality Theorem, which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of $\omega_1$ is preserved by any proper forcing. We…
For a cardinal kappa and a model M of cardinality kappa let No(M) denote the number of non-isomorphic models of cardinality kappa which are L_{infty,kappa}--equivalent to M. In [Sh:133] Shelah established that when kappa is a weakly compact…
W.H. Woodin showed that if $\kappa_1 < \cdots < \kappa_n$ are strong cardinals then two-step ${\bf\Sigma}^1_{n+3}$ generic absoluteness holds after collapsing $2^{2^{\kappa_n}}$ to be countable. We show that this number can be reduced to…
We consider the two-cardinal Kurepa Hypothesis $\mathsf{KH}(\kappa,\lambda)$. We observe that if $\kappa\leq\lambda<\mu$ are infinite cardinals then…
We construct a model of the form $L[A,U]$ that exhibits the simplest structural behavior of $\sigma$-complete ultrafilters in a model of set theory with a single measurable cardinal $\kappa$ , yet satisfies $2^\kappa = \kappa^{++}$. This…
We answer a variant of a question of Rodl and Voigt by showing that, for a given infinite cardinal lambda, there is a graph G of cardinality kappa =(2^lambda)^+ such that for any colouring of the edges of G with lambda colours, there is an…
We study a two-parameter generalization of the Freese-Nation Property of boolean algebras and its order-theoretic and topological consequences. For every regular infinite \kappa, the (\kappa,\kappa)-FN, the (\kappa^+,\kappa)-FN, and the…
We investigate the unbalanced ordinary partition relations of the form $\lambda \rightarrow {(\lambda, \alpha)}^{2}$ for various values of the cardinal $\lambda$ and the ordinal $\alpha$. For example, we show that for every infinite…
We consider global analogues of model-theoretic tree properties. The main objects of study are the invariants related to Shelah's tree property $\kappa_{\text{cdt}}(T)$, $\kappa_{\text{sct}}(T)$, and $\kappa_{\text{inp}}(T)$ and the…