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

Eyal Ackerman, Balázs Keszegh, Günter Rote

Research output: Contribution to journalArticlepeer-review

## Abstract

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon? 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 even, then every pair of sides may cross and so the answer is mn. If exactly one polygon, say the n-gon, has an odd number of sides, it can intersect each side of the m-gon polygon at most n- 1 times; hence there are at most mn- m intersections. It is not hard to construct examples that meet these bounds. 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.

Original language English 1049-1077 29 Discrete and Computational Geometry 68 4 https://doi.org/10.1007/s00454-022-00438-0 Published - Dec 2022

### Bibliographical note

Publisher Copyright:
© 2022, The Author(s).

## Keywords

• Combinatorial geometry
• Ramsey theory
• Simple polygon

## ASJC Scopus subject areas

• Theoretical Computer Science
• Geometry and Topology
• Discrete Mathematics and Combinatorics
• Computational Theory and Mathematics

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