On the ideal $J[\kappa]$

TL;DR

This paper provides a new characterization of the ideal J[κ], confirming the existence of κ-Souslin trees in various models and linking square principles to the existence of such trees under certain cardinal assumptions.

Contribution

It introduces a novel characterization of the ideal J[κ], enabling new existence results for κ-Souslin trees based on square principles and cardinal arithmetic.

Findings

Confirmed existence of κ-Souslin trees in multiple models.

Linked square principles to Souslin tree existence under specific cardinal conditions.

Provided new insights into the structure of the ideal J[κ].

Abstract

Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[\kappa]$, from which we confirm that $\kappa$-Souslin trees exist in various models of interest. As a corollary we get that for every integer $n$ such that $\mathfrak b<2^{\aleph_n}=\aleph_{n+1}$, if $\square(\aleph_{n+1})$ holds, then there exists an $\aleph_{n+1}$-Souslin tree.