# Dimension, Graph and Hypergraph Coloring

DOI: 10.1023/A:1006429830221

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

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## 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*).