Some consequences of reflection on the approachability ideal

We study the approachability ideal $\mathcal{I}[\kappa^+]$ in the context of large cardinals and properties of the regular cardinals below a singular $\kappa$. As a guiding example consider the approachability ideal $\mathcal{I}[\aleph_{\omega+1}]$ assuming that $\aleph_\omega$ is a strong limit. In this case we obtain that club many points in $\aleph_{\omega+1}$ of cofinality $\aleph_n$ for some $n>1$ are approachable assuming the joint reflection of countable families of stationary subsets of $\aleph_n$. This reflection principle holds under $\mathsf{MM}$ for all $n>1$ and for each $n>1$ is equiconsistent with $\aleph_n$ being weakly compact in $L$. This characterizes the structure of the approachability ideal $\mathcal{I}[\aleph_{\omega+1}]$ in models of $\mathsf{MM}$. We also apply our result to show that the Chang conjecture $(\kappa^+,\kappa)\twoheadrightarrow(\aleph_2,\aleph_1)$ fails in models of $\mathsf{MM}$ for all singular cardinals $\kappa$.