Addendum to "on line arrangements in the hyperbolic plane" [European J. Combin.23 (2002) 549-557]

It has been brought to the attention of the authors by János Pach that Theorem 1.1 in the article can be regarded as a reformulation of the main result in [1]. We were not aware of this when preparing our article. In particular, in [1] it is proven that the maximum number of straight line segments connecting n points in convex position in the plane so that no k + 1 of them is crossing is (n 2 ) if n ≤ 2k + 1 and 2nk − (2k+1 2 ) if n ≥ 2k + 1. It is also pointed out that this result is related to problems concerning geometric graphs and is the solution to a problem raised by Bernd Gärtner. The proof of the main result in [1] is somewhat different from our proof of Theorem 1.1. Rather than placing an ordering on line arrangements, elements of a maximum collection of line segments L connecting n points in convex position so that no k + 1 of them is crossing are ranked according to the way in which they cross a particular line segment l ∈ L. This ranking is then used together with an inductive argument on n to bound the number of line segments in L crossing l thus giving the required bound. We thank J. Pach for his comments. J.H. Koolen thanks the Com2MaC-KOSEF for its support. V. Moulton thanks the Swedish Research Council (VR) for its support.