Graphs, Ultrafilters and Colourability
Felix Dilke

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 be the functor from Set to CHaus which maps each discrete set X to its Stone-Cech compactification, the set X of ultrafilters on X. Every graph G with vertex set V naturally gives rise to a graph on the set of ultrafilters on V . In what follows, we interrelate the properties of G and . Perhaps the most striking result is that G can be finitely coloured iff has no loops.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms · Topological and Geometric Data Analysis
