  We discuss ways of adjoining perfect sets of mutually generic random reals.
In particular, we show that if V \sub W are models of ZFC and W contains a
dominating real over V, then W[r], where r is random over W, contains a perfect
tree of mutually random reals over V. This result improves an earlier result by
Bartoszynski and Judah and sheds new light on an old Theorem of Mycielski's.
  We also investigate the existence of perfect free subsets for projective
functions f: (omega^omega)^n \to omega^omega. In particular, we prove that
every projective function has a perfect free subset in the Cohen real model,
while in the Sacks real model the following holds: every Delta^1_2-- function
f: omega^omega \to omega^omega has a perfect free subset and there is a
Delta^1_2--function f: (omega^omega)^2 \to omega^omega without perfect free
subsets. This is connected with recent results of Mildenberger.
  Finally, we show that the existence of a superperfect tree of Cohen reals
over a model V of ZFC implies the existence of a dominating real over V, thus
answering a question addressed by Spinas.
