Partitions of products
Transactions of the American Mathematical Society
This paper extends some applications of a theorem of Halpern and Lauchli on partitions of products of finitary trees. The extensions are to weak infinite products of dense linear orderings, and ultrafilter preservation for finite product Sacks forcing. Introduction. The Halpern-Lauchli (HL) theorem is proved by J. D. Halpern and H. Lauchli . Another proof is given by Halpern  . The original purpose of the theorem was to give a model for set theory in which the Boolean prime ideal theorem
... rime ideal theorem is true while the axiom of choice is false. This was done by J. D. Halpern and A. Levy  . The Laver-Pincus (LP) theorem is a reformulation of the HL theorem due independently to R. Laver (1969, unpublished) and D. Pincus (1974) . In this form the theorem has been applied to finite products of dense order types (Laver, 1969, unpublished), product Sacks forcing (J. Baumgartner, 1974, unpublished), and the analytic Ramsey theory of ww (K. Milliken ). Milliken gives proofs of the Laver-Pincus theorem in  and . He also gives a joint generalization of it and Ramsey's theorem. This paper will strengthen the existing applications of the HL theorem as they pertain to models of set theory without the axiom of choice, products of dense linear orderings, and product Sacks forcing. In the course of the paper we will give two proofs of the LP theorem. The original proof is one which parallels that of the HL theorem. One of our proofs will proceed directly from the result of the HL theorem. The other proof will be along the lines of the original proof but will obtain a stronger result. The paper is organized as follows. The LP theorem and other results on products of trees are stated in tree form in § 1. They are restated in matrix form in §2 and the other tree product theorems are reduced to the matrix form of the LP theorem there. The first proof of the LP theorem, the one proceeding directly from the result of the HL theorem, is given in §3. §4 contains a theorem which is a special case of a conjecture of Pincus  on a model of set theory without the axiom of choice. The results of Halpern , Halpern and Levy  and Pincus  suggest a way to proceed from a given model of set theory without the axiom of choice to a combinatorial theorem which implies that the Boolean prime ideal theorem holds in the model. In  a model is given Received by the editors May 11, 1979 and, in revised form, December 2, 1980 A MS (MOS) subject classifications (1970). Primary 05C55, 04A20, 03E40; Secondary 03E25, 05C05.