  We discuss the relationship between perfect sets of random reals, dominating
reals, and the product of two copies of the random algebra B. Recall that B is
the algebra of Borel sets of 2^omega modulo the null sets. Also given two
models M subseteq N of ZFC, we say that g in omega^omega cap N is a dominating
real over M iff forall f in omega^omega cap M there is m in omega such that
forall n geq m (g(n) > f(n)); and r in 2^omega cap N is random over M iff r
avoids all Borel null sets coded in M iff r is determined by some filter which
is B-generic over M.
  We show that there is a ccc partial order P which adds a perfect set of
random reals without adding a dominating real, thus answering a question asked
by the second author in joint work with T. Bartoszynski and S. Shelah some time
ago. The method of the proof of this result yields also that B times B does not
add a dominating real. By a different argument we show that B times B does not
add a perfect set of random reals (this answers a question that A. Miller asked
during the logic year at MSRI).
