Let \(S, T\) be stationary subsets of an uncountable regular \(\kappa\). Say that \(S <^* T\) iff every stationary subset of \(S\) reflects at almost every point in \(T\). Let \(S^n_j = \{ \alpha < \omega_n : cf(\alpha) = \omega_j \}\). We construct a model with a sequence of stationary sets \(B_n \subseteq S^{n+2}_{n+1}\) for \(n < \omega\), such that \(S^{n+2}_i <^* B_n\) for all \(i \le n\), in which the sets \(B_n\) are as large as possible (it is not possible that \(B_n = S^{n+2}_{n+1}\)).
Joint work with Dorshka Wylie.