# Halpern-Lauchli theorem

In mathematics, the Halpern-Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theorem is true but the axiom of choice is false. It is often called the Halpern-Läuchli theorem, but the proper attribution for the theorem as it is formulated below is to Halpern-Läuchli-Laver-Pincus (HLLP), following (Milliken 1979).

Let d,r &lt; &omega;, $langle T_i: i in d angle$ be a sequence of finitely splitting trees of height &omega;. Let :

Alternatively, let strongly embedded in $T= langle T_i: i in d angle$.

References

#J.D. Halpern and H. Läuchli, A partition theorem, "Trans. Amer. Math. Soc." 124 (1966), 360-367
#Keith R. Milliken, A Ramsey Theorem for Trees, "J. Comb. Theory (Series A)" 26 (1979), 215-237
#Keith R. Milliken, A Partition Theorem for the Infinite Subtrees of a Tree, "Trans. Amer. Math. Soc." 263 No.1 (1981), 137-148
#J.D. Halpern and David Pincus, Partitions of Products, "Trans. Amer. Math. Soc." 267, No.2 (1981), 549-568.

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