### Borel partitions of infinite subtrees of a perfect tree

by Louveau and Velickovic and Shelah. [LVSh:483]

Annals Pure and Applied Logic, 1993

A theorem of Galvin asserts that if the unordered pairs of
reals are partitioned into finitely many Borel classes then there is
a perfect set P such that all pairs from P lie in the same
class. The generalization to n-tuples for n >= 3 is false. Let
us identify the reals with 2^omega ordered by the lexicographical
ordering and define for distinct x,y in 2^omega, D(x,y) to be
the least n such that x(n) not= y(n) . Let the type of an
increasing n-tuple {x_0, ... x_{n-1}}_< be the ordering
<^* on {0, ...,n-2} defined by i<^*j iff D(x_i,x_{i+1})<
D(x_j,x_{j+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. Blass proved an
analogous result is true for any n . 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. We consider extensions of
this result to partitions of infinite increasing sequences of
reals. We show, that for any Borel or even analytic partition of all
increasing sequences of reals there is a perfect set P such that
all strongly increasing sequences from P lie in the same class.

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