On Lihʹs Conjecture concerning Spernerity

Let F be a nonempty collection of subsets of [n] = { 1, 2, …,n}, each having cardinalityt. Denote byPFthe poset consisting of all subsets of [n] which contain at least one member of F, ordered by set-theoretic inclusion. In 1980, K. W. Lih conjectured thatPFhas the Sperner property for all 1 ≤ t ≤ nand every choice of F. This conjecture is known to be true fort = 1 but false, in general, fort ≥ 4. In this paper, we prove Lihʹs conjecture in the caset = 2. e extensive use of fundamental theorems concerning the preservation of Sperner-type properties under direct products of posets.