The Galvin property under the Ultrapower Axiom

We continue the study of the Galvin property. In particular, we deepen the connection between certain diamond-like principles and non-Galvin ultrafilters. We also show that any Dodd sound ultrafilter that is not a $p$-point is non-Galvin. We use these ideas to formulate an essentially optimal large cardinal hypothesis that ensures the existence of a non-Galvin ultrafilter, improving on results of Benhamou and Dobrinen. Finally, we use a strengthening of the Ultrapower Axiom to prove that in all the known canonical inner models, a $\kappa$-complete ultrafilter on $\kappa$ has the Galvin property if and only if it is an iterated sum of $p$-points

Transferring Compactness

We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a cardinal $\kappa$ that is $n$-$d$-stationary for all $n\in \omega$ but not weakly compact. This is in sharp contrast to the situation in the constructible universe $L$, where $\kappa$ being $(n+1)$-$d$-stationary is equivalent to $\kappa$ being $\mathbf{\Pi}^1_n$-indescribable. We also show that it is consistent that there is a cardinal $\kappa\leq 2^\omega$ such that $P_\kappa(\lambda)$ is $n$-stationary for all $\lambda\geq \kappa$ and $n\in \omega$, answering a question of Sakai.Comment: Corrected some typo

A Small Ultrafilter Number at Every Singular Cardinal

We obtain a small ultrafilter number at $\aleph_{\omega_1}$. Moreover, we develop a version of the overlapping strong extender forcing with collapses which can keep the top cardinal $\kappa$ inaccessible. We apply this forcing to construct a model where $\kappa$ is the least inaccessible and $V_\kappa$ is a model of GCH at regulars, failures of SCH at singulars, and the ultrafilter numbers at all singulars are small

Galvin's property at large cardinals and the axiom of determinancy

In the first part of this paper, we explore the possibility for a very large cardinal $\kappa$ to carry a $\kappa$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $\kappa$-complete ultrafilter extends to a non-Galvin one. Oppositely, it is also consistent that every ground model $\kappa$-complete ultrafilter extends to a $P$-point ultrafilter, hence to another one satisfying Galvin's property. We also study Galvin's property at large cardinals in the choiceless context, especially under \textsf{AD}. Finally, we apply this property to a classical pro\-blem in partition calculus by proving the relation $\lambda\rightarrow(\lambda,\omega+1)^2$ under ``\textsf{AD}+$V=L(\mathbb{R})$'' for unboundedly many $\lambda>{\rm cf}(\lambda)>\omega$ below $\Theta$

Kurepa trees and the failure of the Galvin property

We force the existence of a non-trivial $\kappa$-complete ultrafilter over $\kappa$ which fails to satisfy the Galvin property. This answers a question asked by the first author and Moti Gitik

Silver-Free Palladium-Catalyzed C(sp3)H Arylation of Saturated Bicyclic Amine Scaffolds

Herein, we report a silver-free Pd(II)-catalyzed C(sp3)-H arylation of saturated bicyclic and tricyclic amine scaffolds. The reaction provides good yields using a range of aryl iodides and aryl bromides including functionalized examples bearing aldehydes, ketones, esters, free phenols, and heterocycles. The methodology has been applied to medicinally relevant scaffolds. Two of the intermediate palladium complexes in the catalytic cycle have been prepared and characterized, and a mechanism is proposed. Removal of the directing group proceeded with good yield under relatively mild conditions