Metrically Ramsey ultrafilters
Journal Title: Математичні Студії - Year 2018, Vol 49, Issue 2
Abstract
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 χ 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∈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.
Authors and Affiliations
Igor Protasov, Kseniia Protasova
Keywords
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- EP ID EP411715
- DOI 10.15330/ms.49.2.115-121
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