On a problem of Steve Kalikow

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