Some consequences of reflection on the approachability ideal
  We study the approachability ideal I[\kappa^+] in the context of large cardinals properties of the regular cardinals below a singular \kappa. As a guiding example consider the approachability ideal I[\aleph_{\omega+1}] assuming that \aleph_\omega is 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 Martin's maximum 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 I[\aleph_{\omega+1}] in models of Martin's maximum. 
Assaf Sharon, Matteo Viale