Chris Lambie-Hanson

Squares and covering matrices

Abstract Viale introduced covering matrices in his proof that SCH follows from PFA. In the course of the proof and 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 □ κ and □ ( κ + ) for a regular cardinal κ. We provide a detailed picture of the implications between these square principles.

SQUARES AND NARROW SYSTEMS

A narrow system is a combinatorial object introduced by Magidor and Shelah in connection with work on the tree property at successors of singular cardinals. In analogy to the tree property, a cardinal

Simultaneous stationary reflection and square sequences

\kappa

Covering properties and square principles

satisfies the \emph{narrow system property} if every narrow system of height

THE HANF NUMBER FOR AMALGAMATION OF COLORING CLASSES

\kappa

Good and bad points in scales

has a cofinal branch. In this paper, we study connections between the narrow system property, square principles, and forcing axioms. We prove, assuming large cardinals, both that it is consistent that

Bounded stationary reflection II

\aleph_{\omega+1}

Reflection on the Coloring and Chromatic Numbers

satisfies the narrow system property and

Evidence for the microscopic formation of mixed-symmetry states from magnetic moment measurements

\square_{\aleph_{\omega}, < \aleph_{\omega}}

On the strengths and weaknesses of weak squares

holds and that it is consistent that every regular cardinal satisfies the narrow system property. We introduce natural strengthenings of classical square principles and show how they can be used to produce narrow systems with no cofinal branch. Finally, we show that the Proper Forcing Axiom implies that every narrow system of countable width has a cofinal branch but is consistent with the existence of a narrow system of width