The random graph embeds in the curve graph of any infinite genus surface

TL;DR

This paper proves that the random graph can be embedded into the curve graph of an infinite genus surface, demonstrating the surface's curve system is as complex as the random graph's universal structure.

Contribution

It establishes a precise characterization of when the random graph embeds into the curve graph, specifically linking it to the surface's infinite genus.

Findings

Random graph embeds in the curve graph iff the surface has infinite genus

Curve systems on infinite genus surfaces are maximally complex

Provides a new connection between graph theory and surface topology

Abstract

The random graph is an infinite graph with the universal property that any embedding of $G-v$ extends to an embedding of $G$, for any finite graph. In this paper we show that this graph embeds in the curve graph of a surface $\Sigma$ if and only if $\Sigma$ has infinite genus, showing that the curve system on an infinite genus surface is "as complicated as possible".