Abstract
We consider an extension of a question of Erdős on the number of k-gons in a set of n points in the plane. Relaxing the convexity restriction we obtain results on 5-gons and 5-holes (empty 5-gons). In particular, we show a direct relation between the number of non-convex 5-gons and the rectilinear crossing number, provide an improved lower bound for the number of convex 5-holes any point set must contain, and prove that the number of general 5-holes is asymptotically maximized for point sets in convex position.
Research partially supported by the Austrian Science Fund (FWF): P23629-N18 ‘Combinatorial Problems on Geometric Graphs’, and the ESF EUROCORES programme EuroGIGA – CRP ‘ComPoSe’, Austrian Science Fund (FWF): I648-N18.
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Aichholzer, O., Hackl, T., Vogtenhuber, B. (2012). On 5-Gons and 5-Holes. In: Márquez, A., Ramos, P., Urrutia, J. (eds) Computational Geometry. EGC 2011. Lecture Notes in Computer Science, vol 7579. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34191-5_1
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