Dimension, Graph and Hypergraph Coloring

Abstract

There is a natural way to associate with a poset P a hypergraph H P, called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of H P. The ordinary graph G P of incomparable pairs determined by the edges in H P of size 2 can have chromatic number substantially less than H P. We give a new proof of the fact that the dimension of P is 2 if and only if G P is bipartite. We also show that for each t ≥ 2, there exists a poset P t for which the chromatic number of the graph of incomparable pairs of P t is at most 3 t − 4, but the dimension of P t is at least (3 / 2) t − 1. However, it is not known whether there is a function f: N→N so that if P is a poset and the graph of incomparable pairs has chromatic number at most t, then the dimension of P is at most f(t).