  We show that there is always a uniformly antisymmetric f:A-> {0,1} if A
subset R is countable. We prove that the continuum hypothesis is equivalent to
the statement that there is an f:R-> omega with |S_x| <= 1 for every x in R. If
the continuum is at least aleph_n then there exists a point x such that S_x has
at least 2^n-1 elements. We also show that there is a function f:Q-> {0,1,2,3}
such that S_x is always finite, but no such function with finite range on R
exists
