Everywhere unbalanced configurations

Journal article

An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number k such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most k. We give a negative answer to a natural variant of this problem, showing that for every natural number k there exists a finite set of points in the plane together with a pseudoline arrangement such that each pseudoline contains at least two points and there is a pseudoline through any pair of points where the number of points on either side of each pseudoline differ by at least k. Moreover, we may find such a configuration with at most 2^{2^ck} points, which, by a result of Pinchasi, is best possible up to the value of the constant c.

Authors: Conlon, D; Lim, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Advances in Mathematics
• Volume: 480
• Issue: Part A
• Article number: 110445
• Publication date: 2025-07-30
• Acceptance date: 2025-07-09
• DOI: 10.1016/j.aim.2025.110445
• EISSN: 1090-2082
• ISSN: 0001-8708
• Language: English
• Keywords: pseudolines; allowable sequences