# Uncountable dichromatic number without short directed cycles

## TL;DR

This paper proves in ZFC that for any infinite cardinal and any finite cycle length, there exists a large digraph with high dichromatic number avoiding short directed cycles, extending previous consistency results.

## Contribution

It constructs, within ZFC, digraphs of any infinite size and high dichromatic number that lack short directed cycles, answering a question posed by Soukup.

## Key findings

- Existence of large digraphs with high dichromatic number avoiding short cycles.
- Construction works for any infinite cardinal and finite cycle length.
- Answers an open question in the field.

## Abstract

A. Hajnal and P. Erd\H{o}s proved that a graph with uncountable chromatic number cannot avoid short cycles, it must contain for example $ C_4 $ (among other obligatory subgraphs). It was shown recently by D. T. Soukup that, in contrast of the undirected case, it is consistent that for any $ n<\omega $ there exists an uncountably dichromatic digraph without directed cycles shorter than $ n $. He asked if it is provable already in ZFC. We answer his question positively by constructing for every infinite cardinal $ \kappa $ and $ n<\omega $ a digraph of size $ 2^{\kappa} $ with dichromatic number at least $ \kappa^{+} $ which does not contain directed cycles of length less than $ n $ as a subdigraph.