  The Kalikow problem for a pair (lambda, kappa) of cardinal numbers, lambda >
kappa (in particular kappa =2) is whether we can map the family of omega
--sequences from lambda to the family of omega --sequences from kappa in a very
continuous manner. Namely, we demand that for eta, nu in lambda^omega we have:
  eta, nu are almost equal if and only if their images are.
  We show consistency of the negative answer e.g. for aleph_omega but we prove
it for smaller cardinals. We indicate a close connection with the free subset
property and its variants.
