MRP, tree properties and square principles

We show that MRP + MA implies that ITP ( λ,ω 2 ) holds for all cardinal λ ≥ ω 2 . This generalizes a result by Weiβ who showed that PFA implies that ITP ( λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □( λ, ω ) for all λ ≥ ω 2 and we give a direct proof that MRP + MA implies the failure of □( λ, ω 1 ) for all λ ≥ ω 2 .