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).