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.
"MRP, tree properties and square principles." J. Symbolic Logic 76 (4) 1441 - 1452, December 2011. https://doi.org/10.2178/jsl/1318338859