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 + dn6 e), 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.
|Title of host publication||36th International Symposium on Computational Geometry, SoCG 2020|
|Editors||Sergio Cabello, Danny Z. Chen|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|State||Published - 1 Jun 2020|
|Event||36th International Symposium on Computational Geometry, SoCG 2020 - Zurich, Switzerland|
Duration: 23 Jun 2020 → 26 Jun 2020
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||36th International Symposium on Computational Geometry, SoCG 2020|
|Period||23/06/20 → 26/06/20|
Bibliographical noteFunding Information:
Funding 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 and by the National Research, Development and Innovation Office – NKFIH under the grant K 116769.
Eyal Ackerman: The main part of this work was performed during a visit to Freie Universit?t Berlin which was supported by the Freie Universit?t Alumni Program. We thank the reviewers for helpful suggestions.
© Eyal Ackerman, Balázs Keszegh, and Günter Rote; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020).
- Combinatorial geometry
- Ramsey theory
- Simple polygon
ASJC Scopus subject areas