Borel partitions of infinite sequences of reals

The starting point of our work is a Ramsey-type theorem of Galvin (unpublished) which asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes (or even classes which have the property of Baire) then there is a perfect set P such that all pairs from P lie in the same class. The obvious generalization to n-tuples for n ≥ 3 is false. For example, look at the coloring of triples where a triple {x, y, z} with x < y < z is colored red provided that y − x < z − y and blue otherwise. Then any perfect set will contain triples of both colors. Galvin conjectured that this is the only bad thing that can happen. It will be simpler to state this if we identify the reals with 2 ordered by the lexicographical ordering and define for distinct x, y ∈ 2 ∆(x, y) to be the least n such that x(n) 6= y(n). Let the type of an increasing n-tuple of reals {x0, . . . xn−1}< be the ordering ≺ on {0, . . . , n − 2} defined by i ≺ j iff ∆(xi, xi+1) < ∆(xj, xj+1). Galvin proved that for any Borel coloring of triples of reals there is a perfect set P such that the color of any triple from P depends only on its type and conjectured that an analogous result is true for any n. This conjecture has been proved by Blass ([Bl]). As a corollary it follows that if the unordered n-tuples of reals are colored into finitely many Borel classes there is a perfect set P such that the n-tuples from P meet at most (n − 1)! classes. The key ingredient in the proof is the well-known Halpern-Laüchli theorem ([HL]) on partitions of products of finitely many tree. In this paper we consider extensions of this result to partitions of infinite increasing sequences of reals. Define a type of an increasing sequence of reals as before and say that such a sequence {xn : n < ω} is strongly increasing if its type is the standard ordering on ω, i.e. if ∆(xn, xn+1) < ∆(xm, xm+1) whenever n < m. We show, for example, that for any Borel or even analytic partition of all increasing sequences of