William T. Trotter
Department of Mathematics
Arizona State University
Tempe, Arizona 85287-1804 U.S.A.
Report B 98-12
There is a natural way to associate with a poset P a hypergraph HP, called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of HP. The ordinary graph GP of incomparable pairs determined by the edges in HP of size 2 can have chromatic number substantially less that HP. We give a new proof of the fact that the dimensionof P is 2 if and only if GP is bipartite. We also show that for each t > = 2, there exists a poset P for which the chromatic number of the graph of incomparable pairs is t, but the dimension of P is at least (3/2)t-1. However, it is not known whether there is a function f: R -> R 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).
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