On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the following his problems on metrically Ramsey ultrafilters (with hope that somebody can solve them).

A coloring $c:[X]^2\to \{0,1\}$ of the family $[X]^2$ of two-element subsets of a metric space $(X,d)$ is called isometric if there exists a function $\chi:\mathbb R\to \{0,1\}$ such that $c(\{x,y\})=\chi\circ d(x,y)$ for any $\{x,y\}\in [X]^2$.

A free ultrafilter $\mathcal U$ on a metric space $(X,d)$ is called a (metrically) Ramsey if for every (isometric) coloring $c:[X]^2\to\{0,1\}$ there exists a set $U\in\mathcal U$ such that $c{\restriction}[U]^2$ is constant.

Problem 1. Is every metrically Ramsey ultrafilter on the metric space $\mathbb N$ of natural numbers Ramsey?

If not, then

Problem 2. Does there exist a ZFC-example of a metrically Ramsey ultrafilter on $\mathbb N$?

More information on metrically Ramsey ultrafilters can be found in this paper.


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