  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.
