相关论文: Wide gaps with short extenders

The main aim of this paper is to prove that the maximal operator $\sigma_{p}^{\kappa ,\ast }f:=\sup_{n\in \mathbf{P}}\left\vert \sigma_{n}^{\kappa }f\right\vert /\left( n+1\right) ^{1/p-2}$ is bounded from the Hardy space $% H_{p}$ to the…

Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point of a variety). To a countable first-order…

Let $\kappa$ be any regular cardinal. Assuming the existence of a huge cardinal above $\kappa$, we prove the consistency of $\binom{\kappa^{++}}{\kappa^+}\rightarrow\binom{\tau}{\kappa^+}$ for every ordinal $\tau<\kappa^{++}$. Likewise, we…

We address a number of problems on Boolean Algebras. For example, we construct, in ZFC, for any BA B, and cardinal kappa BAs B_1,B_2 extending B such that the depth of the free product of B_1,B_2 over B is strictly larger than the depths of…

We continue [Sh:b, Ch XIII] and [Sh:410]. Let W be an inner model of ZFC. Let kappa be a cardinal in V. We say that kappa-covering holds between V and W iff for all X in V with X subseteq ON and V models |X|< kappa, there exists Y in W such…

Suppose $\kappa$ is $\lambda$-supercompact witnessed by an elementary embedding $j:V\rightarrow M$ with critical point $\kappa$, and further suppose that $F$ is a function from the class of regular cardinals to the class of cardinals…

Suppose that there is a measurable cardinal. If \aleph_\omega is a strong limit cardinal, but the power of \aleph_\omega is bigger than \aleph_{\omega_1}, then there is an inner model with a Woodin cardinal. Modulo the need of the…

Given a function $f : A \to \mathbb{R}^n$ of a certain regularity defined on some open subset $A \subseteq \mathbb{R}^m$, it is a classical problem of analysis to investigate whether the function can be extended to all of $\mathbb{R}^m$ in…

We study higher analogues of the classical independence number on $\omega$. For $\kappa$ regular uncountable, we denote by $i(\kappa)$ the minimal size of a maximal $\kappa$-independent family. We establish ZFC relations between $i(\kappa)$…

We show that if \kappa\ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size \kappa\ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space 2^\kappa\…

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…

Let $X$ be a projective manifold of dimension $n$ and $L$ a strictly nef line bundle on $X$. Then $K_X+tL$ is ample if $t > n+1$ in the following cases. 1.) $\text{dim} X = 3$ unless (possibly) $X$ is a Calabi-Yau with $c_2 \cdot L=0$; 2.)…

Square-kappa-finite, the finite family version of weak square, holds at all cardinals kappa in the Mitchell-Steel inner models.

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erd\H{o}s implying that, for any function $\xi(n)$ tending to infinity with…

Woodin has shown that if there is a measurable Woodin cardinal then there is, in an appropriate sense, a sharp for the Chang model. We produce, in a weaker sense, a sharp for the Chang model using only the existence of a cardinal $\kappa$…

We investigate the circumstances under which there exist a singular cardinal $\mu$ and a short $(\kappa, \mu)$-extender $E$ witnessing "$\kappa$ is $\mu$-strong", such that $\mu$ is singular in $\Ult(V, E)$.

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly kappa+ many normal measures on the least measurable cardinal kappa. This answers a question of Stewart Baldwin. The methods…

Let $n (>3)$ be a prime number and $\Bbb F_{2^n}$ a finite field of $2^n$ elements. Let $L =\Bbb F_{2^n}\cup \{\infty\}$ be the support set and $g(x)$ an irreducible polynomial of degree $6$ over $\Bbb F_{2^n}$. In this paper, we obtain an…

For an inaccessible cardinal $\kappa$, the super tree property (ITP) at $\kappa$ holds if and only if $\kappa$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor…

We present a new version of the Friedman-Magidor theorem: for every measurable cardinal $\kappa$ and $\tau\leq\kappa^{++}$, there exists a forcing extension $V\subseteq V[G]$ such that any normal measure $U\in V$ on $\kappa$ has exactly…