Dimension, Graph and Hypergraph Coloring
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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 than HP. We give a new proof of the fact that the dimension of P is 2 if and only if GP is bipartite. We also show that for each t ≥ 2, there exists a poset Pt for which the chromatic number of the graph of incomparable pairs of Pt is at most 3 t − 4, but the dimension of Pt 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).
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