Strong tree properties, Kurepa trees, and guessing models

TL;DR

This paper explores generalized tree and guessing model properties, introduces a new weakening called the Almost Guessing Property, and demonstrates its implications and consistency results in set theory.

Contribution

It introduces the Almost Guessing Property, providing an alternative formulation of the slender tree property and establishing its consistency in certain models.

Findings

Almost Guessing Property implies failures of square and nonexistence of weak Kurepa trees.

Instances of the Almost Guessing Property hold in the Mitchell model from a strongly compact cardinal.

Constructs a model with stationarily many -guessing models that are not -guessing models.

Abstract

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 prominent combinatorial principles. We introduce a weakening of Viale and Wei\ss's Guessing Model Property, which we call the Almost Guessing Property, and prove that it provides an alternate formulation of the slender tree property in the same way that the Guessing Model Property provides and alternate formulation of the ineffable slender tree property. We show that instances of the Almost Guessing Property have sufficient strength to imply, for example, failures of square or the nonexistence of weak Kurepa trees. We show that these instances of the Almsot Guessing Property hold in the Mitchell model starting from a strongly compact cardinal and prove a number of other consistency results showing that certain implications between the principles under consideration are in general not reversible. In the process, we provide a new answer to a question of Viale by constructing a model in which, for all regular $\theta \geq \omega_2$, there are stationarily many $\omega_2$-guessing models $M \in \mathscr{P}_{\omega_2} H(\theta)$ that are not $\omega_1$-guessing models.