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.
Bibliographical noteFunding Information:
Eyal Ackerman: The main part of this work was performed during a visit to Freie Universität Berlin which was supported by the Freie Universität Alumni Program. Balázs Keszegh: Research supported by the Lendület program of the Hungarian Academy of Sciences (MTA), under the grant LP2017-19/2017, by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by the National Research, Development and Innovation Office—NKFIH under the grant K 116769, K 132696, and FK 132060, and by the ÚNKP-21-5 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund.
© 2022, The Author(s).
- Combinatorial geometry
- Ramsey theory
- Simple polygon
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics