A note on a theorem of Perles concerning non-crossing paths in convex geometric graphs

Abstract

During the last decade a Turán-type result of Perles about the length of the longest non-crossing paths in convex geometric graphs has been receiving some attention in the community studying geometric graphs. In this note we prove that it implies a theorem of Merino, Salazar and Urrutia about the length of the longest alternating paths for a multicoloured point set in convex position. We also give an alternative proof of Perles's theorem based on some ideas from the Merino et al. paper.