Products of Infinitely Many Perfect Trees

Statements (l)-(3) all say the same thing: they are interdeducible. In the 1960s, (2) was noted when pK adds a single Sacks real (that is, when K = 1) and (2) for arbitrary K was asked. Baumgartner (in the early 1970s) derived (2) for K = d < co from HLd; he then raised the question of HL^ and derived (2) from it. Finally, Harrington (in the late 1970s) derived a version of (3), where the functions f{ are from [0, l ] d to [0,1] (d < co), from HLd. Theorems 1 to 3 are concerned with versions and applications of the finite Halpern-Lauchli theorem; HL^ is Theorem 4. Improvements in terms of dense sets and nicely embedded subtrees are given in Theorem 5 and its corollary. Theorems 6 and 7 are slightly improved versions of (2) and (3), respectively. By a perfect subtree of a perfect tree T we shall always mean a downwards closed perfect subtree of T. If T is a tree and t e T, let Tt = {u € T: u ^ r t or t < T u}. If d < co and T = <7]: i < d> is a sequence of trees then an n-dense sequence in f is an X = (Xt: i < d} such that for some m ^ n, each AT, £ 7](m), and for each i and ti £ Tj[n) there is u 6 X{ with t < T. u. If x € (X) 7] then an x-n-dense sequence in f is an n-dense sequence in <(7^)X(: i < d}. The following is a dense set version of the Halpern-Lauchli theorem [7].