Reflecting stationary sets

We prove that the statement “For every pair A, B , stationary subsets of ω 2 , composed of points of cofinality ω , there exists an ordinal α such that both A ∩ α and B ∩ α are stationary subsets of α is equiconsistent with the existence of weakly compact cardinal. (This completes results of Baumgartner and Harrington and Shelah.) We also prove, assuming the existence of infinitely many supercompact cardinals, the statement “Every stationary subset of ω ω+1 has a stationary initial segment.”