If M is a proper class inner model of ZFC and ω2M = ω2, then every sound mouse projecting to ω and not past 0¶ belongs to M. In fact, under the assumption that 0¶ does not belong to M, KM ∥ ω2 is universal for all countable mice in V. Similarly, if M is a proper class inner model of ZFC, δ > ω1 is regular, (δ +)M = δ + and in V there is no proper class inner model with a Woodin cardinal, KM ∥ δ then is universal for all mice in V of cardinality less than δ.

By forcing with Pmax over strong models of determinacy, we obtain models where different square principles at ω2 and ω3 fail. In particular, we obtain a model of 2ℵ0=2ℵ1=ℵ2+¬◻(ω2)+¬◻(ω3).