Diametric Colorings in Ultrametric Spaces

Overview

The classical Ramsey theorem states that for any $k$, there exists $n$ such that any 2-coloring of the edges of $K_n$ contains a monochromatic clique of size $k$. In other words, complete disorder is impossible — structure always emerges in large enough systems.

This work applies the discrete, combinatorial ideas from Ramsey theory to arbitrary ultrametric spaces. More specifically, we color a class of subsets of an ultrametric space $(X, d)$ and show that certain large structures called free ultrafilters contain monochromatic structure. A free ultrafilter $\mathcal{F}$ is a collection of subsets of $X$ closed under intersections and supersets with $\bigcap \mathcal{F} = \emptyset.$ Intuitively, $\mathcal{F}$ is a “large” structure because its sets must overlap, grow upwards, and spread throughout the space.

Let $\Gamma_X$ denote the family of compact subsets of $(X, d)$. A coloring $\chi : \Gamma_X \to [k]$ is diametric if every pair of compact subsets with equal diameters receive the same color. We call a set $M$ monochrome if its compact subsets receive the same color, and we say $\mathcal{F}$ is diametrically Ramsey if every diametric coloring $\chi$ admits a monochrome set $M_\chi \in \mathcal{F}$.

We prove that every infinite ultrametric space contains a diametrically Ramsey ultrafilter.

Theorem. Every infinite ultrametric space contains a sequence $\{x_n\}$ such that any free ultrafilter containing $\{x_n\}$ is diametrically Ramsey.

This extends a result of Protasov and Protasova.