$k$-arrangements of pseudolines and pseudocircles

2026-09-30

AI Summary

This study addresses the inevitability and quantitative growth of simple subarrangements within arrangements of pseudolines and pseudocircles, a fundamental problem in combinatorial geometry. By integrating techniques from combinatorial geometry, graph theory, and Ramsey theory, this work systematically investigates the Ramsey properties of pseudoline arrangements and even-order pseudocircle $k$-arrangements, providing an in-depth analysis of the unavoidable nature of their simple subarrangements. The principal contribution is the proof that, for any fixed $m \ge 3$, the number of unavoidable simple $k$-arrangements grows exponentially with $k$, and that this growth behavior remains independent of the parameter $m$. These findings reveal the intrinsic combinatorial complexity inherent in arrangement structures, offering novel theoretical perspectives for related areas of research.

Abstract

A $k$-arrangement of pseudolines is a set of bi-infinite curves in the plane such that any two of them intersect each other in exactly $k$ points, at which they cross, and it is simple if no three curves meet at a common point. Cyclic arrangements are the only simple $1$-arrangements of pseudolines that are unavoidable, in the Ramsey spirit: for each fixed $m\ge 1$, every sufficiently large simple $1$-arrangement of pseudolines has a cyclic subarrangement of size $m$. We show that, for every $m\ge 3$, the number of unavoidable simple $k$-arrangements of pseudolines of size $m$ grows exponentially with $k$, independently of $m$. For even $k$, we prove an analogous result for $k$-arrangements of pseudocircles.

Problem

• pseudolines
• pseudocircles
• k-arrangements
• Ramsey theory
• unavoidable subarrangements

Innovation

• k-arrangements
• pseudolines
• pseudocircles
• unavoidable arrangements
• Ramsey theory