This is a theorem of Gyula Károlyi, János Pach and Géza Tóth: Ramsey-type results for geometric graphs. I. ACM Symposium on Computational Geometry (Philadelphia, PA, 1996).
Discrete Comput. Geom. 18 (1997), no. 3, 247–255. [Link to preprint](http://dimacs.rutgers.edu/TechnicalReports/abstracts/1995/95-49.html)

In this paper they indeed give an induction proof, but IMHO not an easy one.