On line arrangements in the hyperbolic plane

Abstract

Given a finite collection L of lines in the hyperbolic plane ℍ, we denote by k = k (L) its Karzanov number, i.e., the maximal number of pairwise intersecting lines in L, and by C(L) and n = n (L) the set and the number, respectively, of those points at infinity that are incident with at least one line from L. By using purely combinatorial properties of cyclic sets, it is shown that #L ≤ 2nk - (2k+12) always holds and that #L equals 2nk - (2k+12) if and only if there is no collection L′ of lines in ℍ with L subset of with not equal to sign L′, k(L′) = k(L) and C(L′) = C(L′) = k(L) and C(L′) = C(L). © 2002 Elsevier Science Ltd. All rights reserved.