Squares and Covering Matrices

TL;DR

This paper explores the relationships between various square principles and covering matrices, constructing examples where certain reflection principles fail, thereby advancing understanding of combinatorial set theory.

Contribution

It introduces new constructions of covering matrices using square sequences, revealing failures of reflection principles and analyzing intermediate square principles.

Findings

Constructed covering matrices where CP and S fail.

Mapped implications between intermediate square principles.

Provided new insights into the structure of square principles.

Abstract

Viale introduced covering matrices in his proof that SCH follows from PFA. In the course of the proof amd subsequent work with Sharon, he isolated two reflection principles, CP and S, which, under certain circumstances, are satisfied by all covering matrices of a certain shape. Using square sequences, we construct covering matrices for which CP and S fail. This leads naturally to an investigation of square principles intermediate between $\square_{\kappa}$ and $\square(\kappa^+)$ for a regular cardinal $\kappa$. We provide a detailed picture of the implications between these square principles.