On uniformly antisymmetric functions
Peter Komjath, Saharon Shelah
Abstract
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 |Sx| <= 1 for every x in R. If the continuum is at least alephn then there exists a point x such that Sx has at least 2n-1 elements. We also show that there is a function f:Q-> 0,1,2,3 such that Sx is always finite, but no such function with finite range on R exists
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