Graphs, Ultrafilters and Colourability

TL;DR

This paper explores the relationship between graphs and their ultrafilter-based extensions, showing that a graph is finitely colorable if and only if its ultrafilter extension has no loops.

Contribution

It introduces a novel connection between graph colorability and ultrafilter extensions via the functor to compact Hausdorff spaces.

Findings

G is finitely colorable iff βG has no loops

Ultrafilter extensions preserve certain graph properties

New insights into graph colorability through ultrafilters

Abstract

Let $\beta$ be the functor from Set to CHaus which maps each discrete set X to its Stone-Cech compactification, the set $\beta$ X of ultrafilters on X. Every graph G with vertex set V naturally gives rise to a graph $\beta G$ on the set $\beta V$ of ultrafilters on V . In what follows, we interrelate the properties of G and $\beta G$. Perhaps the most striking result is that G can be finitely coloured iff $\beta G$ has no loops.