Optimal Matrices of Partitions and an Application to Souslin Trees

TL;DR

This paper introduces n-optimal matrices of partitions of natural numbers and applies them to improve decomposition results for strongly homogeneous Souslin trees, addressing questions about their rigidity.

Contribution

It develops the concept of n-optimal matrices of partitions and uses them to enhance the understanding of Souslin trees' structure and rigidity properties.

Findings

Improved decomposition results for strongly homogeneous Souslin trees

New separation results for strong notions of rigidity of Souslin trees

Answering key questions posed by Fuchs and Hamkins

Abstract

The basic result of this note is a statement about the existence of families of partitions of the set of natural numbers with some favourable properties, the n-optimal matrices of partitions. We use this to improve a decomposition result for strongly homogeneous Souslin trees. The latter is in turn applied to separate strong notions of rigidity of Souslin trees, thereby answering a considerable portion of a question of Fuchs and Hamkins.