| Abstrakt: |
Given a metric space (X,d), we say that a mapping χ:[X]² → {0,1} is an isometric coloring if d(x,y) = d(z,t) implies χ({x,y}) = χ({z,t}). A free ultrafilter U on an infinite metric space (X, d) is called metrically Ramsey if, for every isometric coloring x of [X]2, there is a member U ∊ U such that the set [U]² is χ-monochrome. We prove that each infinite ultrametric space (X, d) has a countable subset Y such that each free ultrafilter U on X satisfying Y χ U is metrically Ramsey. On the other hand, it is an open question whether every metrically Ramsey ultrafilter on the natural numbers N with the metric |x -- y| is a Ramsey ultrafilter. We prove that every metrically Ramsey ultrafilter U on N has a member with no arithmetic progression of length 2, and if U has a thin member then there is a mapping f: N → ω such that f(U) is a Ramsey ultrafilter. [ABSTRACT FROM AUTHOR] |
