Let G be the set of all connected graphs on vertex set [n]. Define the partial ordering < on G as follows: for G,H∈G let G<H if E(G)⊂E(H). The poset (G,<) is graded, each level containing the connected graphs with the same number of edges. We prove that (G,<) has the Sperner property, namely that the largest antichain of (G,<) is equal to its largest sized level.
@article{arxiv.1511.08246,
title = {The poset on connected graphs is Sperner},
author = {Stephen G. Z. Smith and István Tomon},
journal= {arXiv preprint arXiv:1511.08246},
year = {2017}
}