Garment numbers of bi-colored point sets in the plane

2026-03-05

AI Summary

This study investigates the existence of five classes of monochromatic four-point geometric configurations in bichromatic point sets in the plane in general position, with a central focus on the long-standing open problem of convex empty monochromatic quadrilaterals. Employing methods from combinatorial geometry and extremal graph theory, the work combines constructive counterexamples with existence proofs to establish tighter upper and lower bounds for each configuration type. The results not only improve upon previously known bounds but also deepen the understanding of the combinatorial properties of bichromatic point sets, offering new insights and partial progress toward resolving the existence question for convex empty monochromatic quadrilaterals.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Applications and emergent phenomena
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We consider colored variants of a class of geometric-combinatorial questions on $k$-gons and empty $k$-gons that have been started around 1935 by Erd\H{o}s and Szekeres. In our setting we have $n$ points in general position in the plane, each one colored either red or blue. A structure on $k$ points is a geometric graph where the edges are spanned by (some of) these points and is called monochromatic if all $k$ points have the same color. Already for $k=4$ there exist interesting open problems. Most prominently, it is still open whether for any sufficiently large bichromatic set there always exists a convex empty, monochromatic quadrilateral. In order to shed more light on the underlying geometry we study the existence of five different monochromatic structures that all use exactly 4 points of a bichromatic point set. We provide several improved lower and upper bounds on the smallest $n$ such that every bichromatic set of at least $n$ points contains (some of) those monochromatic structures.

Problem

• bichromatic point sets
• monochromatic quadrilateral
• empty convex polygon
• Erdős–Szekeres problem
• geometric-combinatorial

Innovation

• bichromatic point sets
• monochromatic structures
• empty convex quadrilaterals
• Erdős–Szekeres-type problems
• geometric combinatorics