A ccc indestructible construction with CH

TL;DR

This paper presents a new construction method using a variant of the Kurepa family to create a ccc indestructible property linked to a complete coherent Suslin tree, impacting partition relations in forcing extensions.

Contribution

It introduces a novel variant of the Kurepa family and uses it to construct a ccc indestructible property associated with Suslin trees, advancing forcing theory.

Findings

Constructs a ccc indestructible property related to Suslin trees.

Shows forcing with the tree induces strong negative partition relations.

Extends understanding of Suslin trees in forcing extensions.

Abstract

We introduce a variant of the Kurepa family. We then use one such family to construct a ccc indestructible property associated with a complete coherent Suslin tree $S$. Moreover, in every ccc forcing extension that preserves Suslin of $S$, forcing with $S$ induces a strong negative partition relation.