Chromatic numbers of directed hypergraphs with no "bad" cycles

TL;DR

This paper investigates the chromatic properties of directed hypergraphs that exclude certain 'bad' cycles, using finite state machine rules, and applies the findings to algebraic structures with bounded width.

Contribution

It introduces a method to construct directed hypergraphs with large chromatic number avoiding 'bad' cycles based on finite state rules, and provides a new proof of the Loop Lemma in a specific algebraic context.

Findings

Constructed directed hypergraphs with arbitrarily large chromatic number avoiding 'bad' cycles.

Applied structural Ramsey theory to analyze cycle properties.

Provided a new proof of the Loop Lemma for bounded width algebras.

Abstract

Imagine that you are handed a rule for determining whether a cycle in a digraph is "good" or "bad", based on which edges of the cycle are traversed in the forward direction and which edges are traversed in the backward direction. Can you then construct a digraph which avoids having any "bad" cycles, but has arbitrarily large chromatic number? We answer this question when the rule is described in terms of a finite state machine. The proof relies on Nesetril and Rodl's structural Ramsey theory of posets with a linear extension. As an application, we give a new proof of the Loop Lemma of Barto, Kozik, and Niven in the special case of bounded width algebras.