We use a variety of combinatorial techniques to prove several theorems concerning fractional dimension of partially ordered sets. In particular, we settle a conjecture of Brightwell and Scheinerman by showing that the fractional dimension of a poser is never more than the maximum degree plus one. Furthermore, when the maximum degree k is at least two, we show that equality holds if and only if one of the components of the poset is isomorphic to Sk + 1, the 'standard example' of a k+ 1-dimensional poset. When w~>3, the fractional dimension of a poset P of width w is less than w unless P contains Sw. If P is a poset containing an antichain A and at most n other points, where n >~ 3, we show that the fractional dimension of P is less than n unless P contains S,. If P contains an antichain A such that all antichains disjoint from A have size at most w ~>4, then the fractional dimension of P is at most 2w, and this bound is best possible.
['Stefan Felsner', 'William T. Trotter']
On the fractional dimension of partially ordered sets