Circle Orders , N - gon Orders and the Crossing Number of Partial Orders

Let F={P1,...,Pm} be a family of sets. A partial order P(F, <) on F is naturally defined by the condition Pi< Pj iff Pi is contained in Pj. When the elements of F are disks (i.e. circles together with their interiors), P(F, <) is called a circle order; if the elements of F are n-polygons, P(F, <) is called an n-gon order. In this paper we study circle orders and n-gon orders. The crossing number of a partial order introduced in [5] is studied here. We show that for every n, there are partial orders with crossing number n. We prove next that the crossing number of circle orders is at most 2 and that the crossing number of n-gon orders is at most 2n. We then produce for every n≥4 partial orders of dimension n which are not circle orders. Also for every n≥3, we prove that there are partial orders of dimension 2n+2 which are not n-gon orders. Finally we prove that every partial order of dimension ≤ 2n is an n-gon order. This research was supported under Natural Sciences and Engineering Research Council of Canada (NSERC Canada) grant numbers A2507 and A0977.