Appears in collections : 13th International workshop in set theory / 13ème Atelier international de théorie des ensembles, 13th International workshop in set theory / 13ème Atelier international de théorie des ensembles

The combinatorics of successors of singular cardinals presents a number of interesting open problems. We discuss the interactions at successors of singular cardinals of two strong combinatorial properties, the stationary set reflection and the tree property. Assuming the consistency of infinitely many supercompact cardinals, we force a model in which both the stationary set reflection and the tree property hold at $\aleph_{\omega^2+1}$. Moreover, we prove that the two principles are independent at this cardinal, indeed assuming the consistency of infinitely many supercompact cardinals it is possible to force a model in which the stationary set reflection holds, but the tree property fails at $\aleph_{\omega^2+1}$. This is a joint work with Menachem Magidor. Keywords : forcing - large cardinals - successors of singular cardinals - stationary reflection - tree property