Related papers: On the ideal $J[\kappa]$

It was conjectured by Flynn that there exists a constant $\kappa$ such that, for any integer $g \ge 2$, any $m \le \kappa g$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its…

From many supercompact cardinals, we show that it is consistent for the tree property to hold at many small successors of singular cardinals, each with a different cofinality. In particular, we construct a model in which the tree property…

We initiate the study of the spectrum $Vspec(\kappa)$ of sets that can be realized as the vanishing levels $V(T)$ of a normal $\kappa$-tree $T$. The latter is an invariant in the sense that if $T$ and $T'$ are club-isomorphic, then the…

Suppose that kappa is a singular cardinal of cofinality omega and GCH holds. Assume that for every n<omega the set of alphas with o(alpha)>= alpha^{+n} is unbounded in kappa.Then there is a cardinal preserving extension satisfying…

We show that the existence of a well-known type of ideals on a regular cardinal $\lambda$ implies a compactness property concerning the specialisability of a tree of height $\lambda$ with no cofinal branches. We also use Neeman's method of…

We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam's theorem and its extension by Hajnal,…

Given an ordinal delta <= lambda and a cardinal theta <= kappa, an ideal J on P_kappa(lambda) is said to be [delta]^{<theta}-normal if given B_e in J for e in P_theta(delta), the set of all a in P_kappa(lambda) such that a in B_e for some e…

Assume $\kappa = \kappa^{< \kappa}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{\kappa>}{\rm Ord}$ and not collapsing cardinals along a linear order $L$. A sufficient condition for this,…

Using deformation theory, Braverman and Joseph obtained an alternative characterisation of the Joseph ideal for simple Lie algebras, which included even type A. In this note we extend that characterisation to define a remarkable quadratic…

We prove the existence and uniqueness of multiple SLE$_\kappa$ associated with any given link pattern for $\kappa\in (4,6]$. We also have the uniqueness for $\kappa\in (6,8)$. The multiple SLE$_\kappa$ law is constructed by first…

We prove that there are infinitely many integers $n$ such that the total number of prime factors of $(n+h_{1})(n+h_{2})...(n+h_{\kappa})$ is at most $(1/2)\kappa\log\kappa+O(\kappa)$, provided $\kappa$ is sufficiently large.

We study the approachability ideal I[\kappa^+] in the context of large cardinals properties of the regular cardinals below a singular \kappa. As a guiding example consider the approachability ideal I[\aleph_{\omega+1}] assuming that…

We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple method…

We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $\kappa$, updating some classical constructions in the process. This includes models of…

Motivated by the goal of constructing a model in which there are no $\kappa$-Aronszajn trees for any regular $\kappa>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square…

A cardinal kappa is countably closed if mu^omega < kappa whenever mu < kappa. Assume that there is no inner model with a Woodin cardinal and that every set has a sharp. Let K be the core model. Assume that kappa is a countably closed…

In 1984, Ditor asked two questions: (1) For each $n\in\omega$ and infinite cardinal $\kappa$, is there a join-semilattice of breadth $n+1$ and cardinality $\kappa^{+n}$ whose principal ideals have cardinality $< \kappa$? (2) For each $n \in…

If T is an iteration tree on K and F is a countably certified extender that coheres with the final model of T, then F is on the extender sequence of the final model of T. Several applications of maximality are proved, including: o K…

The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa$ is…

We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of $\square(\kappa)$ introduced by Brodsky and Rinot for the purpose of constructing $\kappa$-Souslin…