Felsner, S. & Trotter, W.T. Order (2000) 17: 167. doi:10.1023/A:1006429830221

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).