On the ideal $J[\kappa ]$

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.