The random graph embeds in the curve graph of any infinite genus surface
  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". 
Edgar A. Bering IV and Jonah Gaster
math.GT