On the fractional dimension of partially ordered sets Original Research Article

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 poset 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 Sn. 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.