The unavoidable arrangements of pseudocircles

TL;DR

This paper extends the concept of unavoidable arrangements from pseudolines to pseudocircles, demonstrating that three specific arrangements of pseudocircles are unavoidable in large simple arrangements.

Contribution

It identifies three unavoidable arrangements of pseudocircles, generalizing known results from pseudolines to pseudocircles.

Findings

Three unavoidable arrangements of pseudocircles identified

Large simple arrangements always contain one of these arrangements

Extends classical pseudoline arrangement results to pseudocircles

Abstract

It is known that cyclic arrangements are the only {\em unavoidable} simple arrangements of pseudolines: for each fixed $m\ge 1$, every sufficiently large simple arrangement of pseudolines has a cyclic subarrangement of size $m$. In the same spirit, we show that there are three unavoidable arrangements of pseudocircles.