Squares, ascent paths, and chain conditions

TL;DR

This paper explores the relationships between square principles, ascent paths, and chain conditions in trees, establishing new consistency results and constructing models to analyze their interactions and implications.

Contribution

It demonstrates that square principles imply the existence of certain Aronszajn trees with ascent paths and provides a comprehensive analysis of their consistency strengths.

Findings

Square(5) implies existence of 5-Aronszajn trees with 5-ascent paths.

Constructed a model with 5_2-Aronszajn trees all containing 5_0-ascent paths.

Showed that certain chain conditions imply the failure of square(5).

Abstract

With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todor\v{c}evi\'{c}'s principle $\square(\kappa)$ implies an indexed version of $\square(\kappa,\lambda)$, we show that for all infinite, regular cardinals $\lambda<\kappa$, the principle $\square(\kappa)$ implies the existence of a $\kappa$-Aronszajn tree containing a $\lambda$-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which $\aleph_2$-Aronszajn trees exist and all such trees contain $\aleph_0$-ascent paths. Finally, we use our techniques to show that the assumption that the $\kappa$-Knaster property is countably productive and the assumption that every $\kappa$-Knaster partial order is $\kappa$-stationarily layered both imply the failure of $\square(\kappa)$.