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# An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

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LIPIcs.SoCG.2020.1.pdf
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• 18 pages

## Acknowledgements

We thank the reviewers for helpful suggestions.

## Cite As

Eyal Ackerman, Balázs Keszegh, and Günter Rote. An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.1

## Abstract

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn-(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn-(m + ⌈ n/6 ⌉), for m ≥ n. We prove a new upper bound of mn-(m+n)+C for some constant C, which is optimal apart from the value of C.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
• Mathematics of computing → Combinatoric problems
##### Keywords
• Simple polygon
• Ramsey theory
• combinatorial geometry

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## References

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2. Michael B. Dillencourt, David M. Mount, and Alan Saalfeld. On the maximum number of intersections of two polyhedra in 2 and 3 dimensions. In Proceedings of the 5th Canadian Conference on Computational Geometry, Waterloo, Ontario, Canada, August 1993, pages 49-54. University of Waterloo, 1993.
3. P. Erdős and L. Moser. On the representation of directed graphs as unions of orderings. Magyar Tud. Akad. Mat. Kutató Int. Közl., 9:125-132, 1964.
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5. Felix Günther. The maximum number of intersections of two polygons, July 2012. withdrawn by the author. URL: http://arxiv.org/abs/1207.0996.
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