Proxy principles in combinatorial set theory

TL;DR

This paper comprehensively surveys the proxy principles in combinatorial set theory, highlighting their utility in constructing Souslin trees with complex features and presenting new results on their applications in tree construction.

Contribution

It provides a detailed exposition of proxy principles, surveys their applications, and introduces new constructions of Souslin trees with specific properties related to derived trees.

Findings

Proxy principles facilitate the construction of complex Souslin trees.

Existence of Souslin trees with all n-derived trees Souslin but (n+1)-power special.

Rich spectrum of models where proxy principles hold.

Abstract

The parameterized proxy principles were introduced by Brodsky and Rinot in a 2017 paper, as new foundations for the construction of $\kappa$-Souslin trees in a uniform way that does not depend on the nature of the (regular uncountable) cardinal $\kappa$. Since their introduction, these principles have facilitated construction of Souslin trees with complex combinations of features, and have enabled the discovery of completely new scenarios in which Souslin trees must exist. Furthermore, the proxy principles have found new applications beyond the construction of trees. This paper opens with a comprehensive exposition of the proxy principles. We motivate their very definition, emphasizing the utility of each of the parameters and the consequent flexibility that they provide. We then survey the findings surrounding them, presenting a rich spectrum of unrelated models and configurations in which the proxy principles are known to hold, and showcasing a gallery of Souslin trees constructed from the principles. The last two sections of the paper offer new results. In particular, for every positive integer $n$, we give a construction of a $\lambda^+$-Souslin tree all of whose $n$-derived trees are Souslin, but whose $(n+1)$-power is special.