The unavoidable rotation systems

TL;DR

This paper generalizes a known result about large topological complete graphs containing specific subgraphs to the broader context of abstract rotation systems, expanding understanding of their structural properties.

Contribution

It extends the theorem of Pach, Solymosi, and Tóth from topological graphs to abstract rotation systems, broadening the scope of subgraph containment results.

Findings

Established that large abstract rotation systems necessarily contain certain canonical substructures.

Generalized existing topological graph results to a more abstract combinatorial setting.

Provided new insights into the structure of large rotation systems.

Abstract

For each positive integer $m$, Pach, Solymosi, and T\'oth identified two canonical complete topological subgraphs $C_m$ and $T_m$, and proved that every sufficiently large topological complete graph contains $C_m$ or $T_m$ as a subgraph. We generalize this result in the setting of abstract rotation systems.