Several variants of the Halpern-Läuchli Theorem for trees of uncountable height are investigated. For κ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one tree on any infinite cardinal. For any finite d ≥ 2, we prove the consistency of the Halpern-Läuchli Theorem on d many normal κ-trees at a measurable cardinal κ, given the consistency of a κ + d-strong cardinal. This follows from a more general consistency result at measurable κ, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.

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Given sets X,Y$X,Y$ and a regular cardinal μ, let normalΦfalse(X,Y,μfalse)$\Phi (X,Y,\mu )$ be the statement that for any function f:X×Y→μ$f : X \times Y \rightarrow \mu$, there are functions g1:X→μ$g_1 : X \rightarrow \mu$ and g2:Y→μ$g_2 : Y \rightarrow \mu$ such that for all false(x,yfalse)∈X×Y$(x,y) \in X \times Y$, f(x,y)≤maxfalse{g1(x),g2(y)false}$f(x,y) \le \max \lbrace g_1(x), g_2(y) \rbrace$. In ZFC$\mathsf {ZFC}$, the statement normalΦfalse(ω1,ω1,ωfalse)$\Phi (\omega _1, \omega _1, \omega )$ is false. However, we show the theory sans-serifZF+“the4.ptclub4.ptfilter4.pton4.ptω14.ptis4.ptnormal”+normalΦfalse(ω1,ω1,ωfalse)$\mathsf {ZF}+ \text{``the club filter on $\omega _1$ is normal''} + \Phi (\omega _1, \omega _1, \omega )$ (which is implied by sans-serifZF+sans-serifDC$\mathsf {ZF}+ \mathsf {DC}$+ “V=Lfalse(double-struckRfalse)$V = L(\mathbb {R})$” + “ω1 is measurable”) implies that for every α<ω1$\alpha < \omega _1$ there is a κ∈false(α,ω1false)$\kappa \in (\alpha ,\omega _1)$ such that in some inner model, κ is measurable with Mitchell order ≥α$\ge \alpha$.

Abstract. For each a ∈ ω ω, we define a Baire class one function f a : ω ω → ω ω which encodes a in a certain sense. We show that for each Borel g :1 (c) where c is any code for g. We generalize this theorem for g in larger pointclasses Γ., then a ∈ M 1+n (c).