Covering properties and square principles

TL;DR

This paper explores the relationships between covering matrices, reflection principles, and square principles, demonstrating consistency results and limitations under large cardinal assumptions and stationary reflection hypotheses.

Contribution

It establishes new consistency results linking square principles with covering matrix reflection principles and analyzes the limitations imposed by cardinal arithmetic and reflection hypotheses.

Findings

Consistency of a(, 2) with (, ) for all  with ^+ <  under large cardinal assumptions.

Construction of - covering matrices that fail certain reflection principles.

Limits on the existence of normal -covering matrices imposed by cardinal arithmetic and stationary reflection.

Abstract

Covering matrices were introduced by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of his work and in subsequent work with Sharon, he isolated two reflection principles, $\mathrm{CP}$ and $\mathrm{S}$, which may hold of covering matrices. In this paper, we continue previous work of the author investigating connections between failures of $\mathrm{CP}$ and $\mathrm{S}$ and variations on Jensen's square principle. We prove that, for a regular cardinal $\lambda > \omega_1$, assuming large cardinals, $\square(\lambda, 2)$ is consistent with $\mathrm{CP}(\lambda, \theta)$ for all $\theta$ with $\theta^+ < \lambda$. We demonstrate how to force nice $\theta$-covering matrices for $\lambda$ which fail to satisfy $\mathrm{CP}$ and $\mathrm{S}$. We investigate normal covering matrices, showing that, for a regular uncountable $\kappa$, $\square_\kappa$ implies the existence of a normal $\omega$-covering matrix for $\kappa^+$ but that cardinal arithmetic imposes limits on the existence of a normal $\theta$-covering matrix for $\kappa^+$ when $\theta$ is uncountable. We also investigate certain increasing sequences of functions which arise from covering matrices and from PCF-theoretic considerations and show that a stationary reflection hypothesis places limits on the behavior of these sequences.