A theorem of Todorcevic on universal Baireness

Given a cardinal κ, a set A ⊂ ωω is κ-universally Baire if there exist trees S, T which project to A and its complement respectively, and which continue to project to complements after the forcing Coll(ω, κ). Todor-cevic [3] has shown that under Martin’s Maximum every set of reals of cardinality ℵ1 is ℵ1-universally Baire but not ℵ2-universally Baire. Here we present alternate proofs of these two facts. Lemma 0.1. Suppose that S is a tree on ω×γ for some ordinal γ, and suppose that some forcing P adds a new real to the projection of S. Then the projection of S contains a perfect set in the ground model. Proof. Let X be a countable elementary submodel of H((2|P |)+) containing P and S and a name τ for a new element of the projection of S. Then there exists a perfect set of filters contained in X ∩ P deciding a perfect set of values for τ along with element of γω witnessing that these realizations of τ are in the projection of S. Then all of these realizations of τ are in the projection of S. Recall that Martin’s Maximum implies that the nonstationary ideal on ω1 (NSω1) is precipitous, and that 2 ℵ1 = ℵ2 [1]. Theorem 0.2. Suppose that NSω1 is precipitous. Let A be an uncountable set of reals not containing a perfect set. Then A is not 2ℵ1-universally Baire. Proof. Let A be an uncountable set of reals with no perfect subset, and towards a contradiction let S, T be trees witnessing that A is 2ℵ1-universally Baire. Since A has no perfect subset, the projection of S in the collapse extension has to be exactly A. Pick an ω1-sequence a ̄ of reals from A. Let G ⊂ P(ω1)/NSω1 be a V-generic filter, and let j: V →M be the corresponding embedding. Since A is 2ℵ1-universally Baire, the ωV1-st element x of j(ā) is in the projection of T, and so it is in the projection of j(T). Furthermore, since M is wellfounded, M | = x ∈ p[j(T)]. Then in V stationarily many members of a ̄ are in the projection of T, giving a contradiction