Skip to Main Content
HBS Home
  • About
  • Academic Programs
  • Alumni
  • Faculty & Research
  • Baker Library
  • Giving
  • Harvard Business Review
  • Initiatives
  • News
  • Recruit
  • Map / Directions
Faculty & Research
  • Faculty
  • Research
  • Featured Topics
  • Academic Units
  • …→
  • Harvard Business School→
  • Faculty & Research→
Publications
Publications
  • Article
  • Graphs and Combinatorics

Every Large Point Set Contains Many Collinear Points or an Empty Pentagon

By: Zachary Abel, Brad Ballinger, Prosenjit Bose, Sébastien Collette, Vida Dujmović, Ferran Hurtado, Scott Duke Kominers, Stefan Langerman, Attila Pór and David Wood
  • Format:Print
ShareBar

Abstract

We prove the following generalised empty pentagon theorem for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique” conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005].

Keywords

Erdős-Szekeres Theorem; Happy End Problem; Big Line Or Big Clique Conjecture; Empty Quadrilateral; Empty Pentagon; Empty Hexagon

Citation

Abel, Zachary, Brad Ballinger, Prosenjit Bose, Sébastien Collette, Vida Dujmović, Ferran Hurtado, Scott Duke Kominers, Stefan Langerman, Attila Pór, and David Wood. "Every Large Point Set Contains Many Collinear Points or an Empty Pentagon." Graphs and Combinatorics 27, no. 1 (January 2011): 47–60.
  • Find it at Harvard
  • Purchase

About The Author

Scott Duke Kominers

Entrepreneurial Management
→More Publications

More from the Authors

    • May 2023
    • Faculty Research

    Twitter Blues: Does Paid Verification Check Out?

    By: Scott Duke Kominers, Das Narayandas and Kerry Herman
    • April 2023
    • Faculty Research

    Bored Ape Yacht Club: Navigating the NFT World (A)

    By: Das Narayandas, Scott Duke Kominers and Kerry Herman
    • April 6, 2023
    • a16zcrypto.com

    A New NFT Launch Strategy: The Wave Mint

    By: Scott Duke Kominers and 1337 Skulls Sers
More from the Authors
  • Twitter Blues: Does Paid Verification Check Out? By: Scott Duke Kominers, Das Narayandas and Kerry Herman
  • Bored Ape Yacht Club: Navigating the NFT World (A) By: Das Narayandas, Scott Duke Kominers and Kerry Herman
  • A New NFT Launch Strategy: The Wave Mint By: Scott Duke Kominers and 1337 Skulls Sers
ǁ
Campus Map
Harvard Business School
Soldiers Field
Boston, MA 02163
→Map & Directions
→More Contact Information
  • Make a Gift
  • Site Map
  • Jobs
  • Harvard University
  • Trademarks
  • Policies
  • Accessibility
  • Digital Accessibility
Copyright © President & Fellows of Harvard College