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