On the fractional dimension of partially ordered sets

Abstract. 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 S k+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 S w. 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 n. 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. Mathematics Subject Classication (1991). 06A07, 05C35. Key words. Partially poset, dimension, fractional dimension, degree