Authors: Oswin Aichholzer ¹; Jean Cardinal ²; Thomas Hackl ¹; Ferran Hurtado ³; Matias Korman ^4,5,6; Alexander Pilz ¹; Rodrigo I. Silveira ^3,7; Ryuhei Uehara ⁸; Pavel Valtr ^9,10; Birgit Vogtenhuber ¹; Emo Welzl ¹¹

• 1 Graz University of Technology [Graz]
• 2 Université libre de Bruxelles
• 3 Universitat Politècnica de Catalunya [Barcelona]
• 4 National Institute of Informatics
• 5 Japan Science and Technology Agency
• 6 IRATO / Kawarabayashi Large Graph Project [Japan]
• 7 Centro de Investigaçao e Desenvolvimento em Matematica e Aplicaçoes
• 8 Japan Advanced Institute of Science and Technology
• 9 Charles University [Prague]
• 10 Univerzita Karlova [Praha, Česká republika] = Charles University [Prague, Czech Republic] [UK]
• 11 Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich]

We prove that the dual graph of any arrangement of n lines in general position always contains a path of length at least n2/4. Further, we show that in every arrangement of n red and blue lines — in general position and not all of the same color — there is a simple path through at least n cells where red and blue lines are crossed alternatingly.