The poset on connected graphs is Sperner

Let C be the set of all connected graphs on vertex set [n]. Then C is endowed with the following natural partial ordering: for G,H∈C, let G≤H if G is a subgraph of H. The poset (C,≤) is graded, each level containing the connected graphs with the same number of edges. We prove that (C,≤) has the Sperner property, namely that the largest antichain of (C,≤) is equal to its largest sized level. This answers a question of Katona.