A LOWER BOUND ON THE AREA OF A 3-COLOURED DISK PACKING

Given a set of unit disks in the plane with union area A, what fraction of A can be covered by selecting a pairwise disjoint subset of the disks? Rado conjectured 1/4 and proved 1/4.41. Motivated by the problem of channel assignment for wireless access points, in which use of 3 channels is a standard practice, we consider a variant where the selected subset of disks must be 3-colourable with disks of the same colour pairwise-disjoint. For this variant of the problem, we conjecture that it is always possible to cover at least 1/1.41 of the union area and prove 1/2.09. We also provide an O(n2) algorithm to select a subset achieving a 1/2.77 bound. Finally, we discuss some results for other numbers of colours.

Free i-Groups and vector lattices

The purpose of this paper is to present three somewhat disparate results on free objects in three different classes of λ-groups. The first is that no proper ideal of a finitely generated free vector lattice can itself be a free vector lattice. Second, each free abelian lgroup is characteristically simple. The third result is that each disjoint subset of a free (non-abelian) lgroup is countable.

The hulls of representable l-groups and f-rings

A lattice-ordered group (“l-group”) G will be calleda P-group if G = g″ ⊕ g′ for each g ∈ G (projectable)an SP-group if G = C ⊕ C′ for each polar C of G (strongly projectable)an L-group if each disjoint subset has a 1. u. b. (laterally complete)an O group if it is both an L-group and a P-group (orthocomplete).G is representable if it is an l-subgroup of a cardinal product of totally ordered groups. It follows that a P-group must be representable and hence SP-groups and O-groups are also representable.

Lifting Disjoint Sets in Vector Lattices

A subset {s α| α ϵ △} of a lattice-ordered group (l-group) is disjoint if Sα Λ Sβ= 0 for all α≠ β in △. An l-group G has the lifting property if for each l-ideal S of G and each countable disjoint subset X1, X2, … of G/S one can choose elements 0 ≦ xi ϵ Xi so that x1, x2, … is a disjoint subset of G. In (2) Topping showed by an example, that uncountable sets of disjoint elements cannot necessarily be lifted and asserted (Theorem 8) that each vector lattice has the lifting property. His proof is valid for finite disjoint subsets of G/S, but we show by an example that this is, in general, all that one can establish.