Orientations of graphs with uncountable chromatic number

TL;DR

This paper investigates digraphs with uncountable dichromatic number and orientations of graphs with uncountable chromatic number, revealing surprising contrasts and independence results within set theory.

Contribution

It proves the existence of digraphs with uncountable dichromatic number and large digirth consistently, and shows certain graphs admit orientations with uncountable dichromatic number in ZFC, while some statements are independent of ZFC.

Findings

Existence of digraphs with uncountable dichromatic number and arbitrarily large digirth.

Certain well-known graphs admit orientations with uncountable dichromatic number in ZFC.

Some statements about orientations with uncountable dichromatic number are independent of ZFC.

Abstract

Motivated by an old conjecture of P. Erd\H{o}s and V. Neumann-Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable chromatic number if its vertices cannot be covered by countably many independent sets, and a digraph has uncountable dichromatic number if its vertices cannot be covered by countably many acyclic sets. We prove that consistently there are digraphs with uncountable dichromatic number and arbitrarily large digirth; this is in surprising contrast with the undirected case: any graph with uncountable chromatic number contains a 4-cycle. Next, we prove that several well known graphs (uncountable complete graphs, certain comparability graphs, and shift graphs) admit orientations with uncountable dichromatic number in ZFC. However, we show that the statement "every graph $G$ of size and chromatic number $\omega_1$ has an orientation $D$ with uncountable dichromatic number" is independent of ZFC.