Spencer Unger

Aronszajn trees and the successors of a singular cardinal

From large cardinals we obtain the consistency of the existence of a singular cardinal κ of cofinality ω at which the Singular Cardinals Hypothesis fails, there is a bad scale at κ and κ++ has the tree property. In particular this model has no special κ+-trees.

Fragility and indestructibility of the tree property

We prove various theorems about the preservation and destruction of the tree property at ω2. Working in a model of Mitchell [9] where the tree property holds at ω2, we prove that ω2 still has the tree property after ccc forcing of size

Combinatorics at ℵω

{\aleph_1}

A model of Cummings and Foreman revisited

or adding an arbitrary number of Cohen reals. We show that there is a relatively mild forcing in this same model which destroys the tree property. Finally we prove from a supercompact cardinal that the tree property at ω2 can be indestructible under ω2-directed closed forcing.

Homogeneous changes in cofinalities with applications to HOD

Abstract We show that every locally finite bipartite Borel graph satisfying a strengthening of Halls condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theorem of Dougherty and Foreman who showed that there is a paradoxical decomposition of the unit ball in R 3 using Baire measurable pieces. We also obtain a Baire category solution to the dynamical von Neumann–Day problem: if a is a nonamenable action of a group on a Polish space X by Borel automorphisms, then there is a free Baire measurable action of F 2 on X which is Lipschitz with respect to a.

The strong tree property and weak square

Abstract We construct a model in which the singular cardinal hypothesis fails at ℵ ω . We use characterizations of genericity to show the existence of a projection between different Prikry type forcings.

Borel circle squaring

Abstract We improve the best known result on successive regular cardinals with the tree property. In particular we prove that relative to an increasing ω + ω -sequence of supercompact cardinals it is consistent that every regular cardinal on the interval [ ℵ 2 , ℵ ω ⋅ 2 ) has the tree property.

MODIFIED EXTENDER BASED FORCING

Abstract This paper concerns the model of Cummings and Foreman where from ω supercompact cardinals they obtain the tree property at each ℵ n for 2 ≤ n ω . We prove some structural facts about this model. We show that the combinatorics at ℵ ω + 1 in this model depend strongly on the properties of ω 1 in the ground model. From different ground models for the Cummings–Foreman iteration we can obtain either ℵ ω + 1 ∈ I [ ℵ ω + 1 ] and every stationary subset of ℵ ω + 1 reflects or there are a bad scale at ℵ ω and a non-reflecting stationary subset of ℵ ω + 1 ∩ cof ( ω 1 ) . We also prove that regardless of the ground model a strong generalization of the tree property holds at each ℵ n for n ≥ 2 .