The poset on connected graphs is Sperner

Abstract

Let $\mathcal{G}$ be the set of all connected graphs on vertex set $[n]$. Define the partial ordering $<$ on $\mathcal{G}$ as follows: for $G,H\in \mathcal{G}$ let $G<H$ if $E(G)\subset E(H)$. The poset $(\mathcal{G},<)$ is graded, each level containing the connected graphs with the same number of edges. We prove that $(\mathcal{G},<)$ has the Sperner property, namely that the largest antichain of $(\mathcal{G},<)$ is equal to its largest sized level.