Nice labeling problem for event structures: a counterexample

TL;DR

This paper provides a counterexample to a long-standing conjecture in event structure theory, showing that not all finite degree structures can be labeled with finitely many labels, challenging previous assumptions.

Contribution

The paper constructs a counterexample to Rozoy and Thiagarajan's conjecture using Burling's hypergraph construction and median graph theory, disproving the conjecture.

Findings

Counterexample disproves the conjecture.

Finite degree event structures may require infinitely many labels.

The result impacts the understanding of event structure labelings.

Abstract

In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (called also the nice labeling problem) asserting that any (coherent) event structure with finite degree admits a labeling with a finite number of labels, or equivalently, that there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that an event structure with degree $\le n$ admits a labeling with at most $f(n)$ labels. Our counterexample is based on the Burling's construction from 1965 of 3-dimensional box hypergraphs with clique number 2 and arbitrarily large chromatic numbers and the bijection between domains of event structures and median graphs established by Barth\'elemy and Constantin in 1993.