Circle orders, N-gon orders and the crossing number

Abstract

Let Φ={P ₁,...,P _m } be a family of sets. A partial order P(Φ, <) on Φ is naturally defined by the condition P _i <P _j iff P _i is contained in P _j . When the elements of Φ are disks (i.e. circles together with their interiors), P(Φ, <) is called a circle order; if the elements of Φ are n-polygons, P(Φ, <) 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.