On Lih's conjecture concerning Spernerity
\textit{Ko-Wei Lih} proved in 1980 [J. Comb. Theory, Ser. A 29, 182-185 (1980; Zbl 0446.05002)] that if \(Y\) is a fixed \(k\)-element subset of \([n]\) and \(P\) denotes the poset that consists of all subsets of \([n]\) having non-empty intersection with \(Y\) then \(P\) satisfies the Sperner property. Lih observed that \(P\) can be considered as a filter in the Boolean algebra \({\mathcal B}_n\) generated by \(k\) one-element sets. From this result he conjectured that if the poset \(P\) is a filter in \({\mathcal B}_n\) generated by a non-empty collection of \(t\)-element subsets, then \(P\) is also Sperner. The conjecture was proved to be false for \(t\geq 4\). This paper proves it for the case \(t=2\).
scientific article; zbMATH DE number 1275379