Abstract. We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(ω 1 ) which is Δ 1 definable with parameter a subset of ω 1 . Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes ℵ 2 and also satisfies BPFA must contain all subsets of ω 1 . We show as applications how to build minimal models of BPFA and that BPFA implies that the decision problem for the Härtig quantifier is not lightface projective.

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, on the occasion of his birthday.Abstract. We study the topological version of the partition calculus in the setting of countable ordinals. Let α and β be ordinals and let k be a positive integer. We write β →top (α, k) 2 to mean that, for every red-blue coloring of the collection of 2-sized subsets of β, there is either a red-homogeneous set homeomorphic to α or a blue-homogeneous set of size k. The least such β is the topological Ramsey number R top (α, k).We prove a topological version of the Erdős-Milner theorem, namely that R top (α, k) is countable whenever α is countable. More precisely, we prove that R top (ω ω β , k + 1) ≤ ω ω β·k for all countable ordinals β and finite k. Our proof is modeled on a new easy proof of a weak version of the Erdős-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of α, proving among other results thatOur computations use a variety of techniques, including a topological pigeonhole principle for ordinals, considerations of a tree ordering based on the Cantor normal form of ordinals, and some ultrafilter arguments.

If the bounded proper forcing axiom BPFA holds and ω1 = ω1L, then there is a lightface Σ31 well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of “David's trick.” We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface Σ41 for many “consistently locally certified” relations R on ℝ. This is accomplished through a use of David's trick and a coding through the Σ2 stable ordinals of L.

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