MRP, tree properties and square principles

Abstract

We show that MRP+MA implies that ITP(λ,ω₂) holds for all cardinal λ ≥ ω₂. This generalizes a result by Weiß who showed that PFA implies that ITP(λ, ω₂) holds for all cardinal λ ≥ ω₂. 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≥ ω₂ and we give a direct proof that MRP+MA implies the failure of □(λ,ω₁) for all λ≥ω₂.