Sperner theorems for unrelated copies of some partially ordered sets in a powerset lattice and minimum generating sets of powers of distributive lattices

Abstract

For a finite poset (partially ordered set) $U$ and a natural number $n$, let Sp$(U,n)$ denote the largest number of pairwise unrelated copies of $U$ in the powerset lattice (AKA subset lattice) of an $n$-element set. If $U$ is the singleton poset, then Sp$(U,n)$ was determined by E. Sperner in 1928; this result is well known in extremal combinatorics. Later, exactly or asymptotically, Sperner's theorem was extended to other posets by A. P. Dove, J. R. Griggs, G. O. H. Katona, D Nagy, J. Stahl, and W. T. Jr. Trotter. We determine Sp$(U,n)$ for all finite posets with 0 and 1, and we give reasonable estimates for the ``V-shaped'' 3-element poset and the 4-element poset with 0 and three maximal elements. For a lattice $L$, let Gmin($L$) denote the minimum size of generating sets of $L$. We prove that if $U$ is the poset of the join-irreducible elements of a finite distributive lattice $D$, then the function $k\mapsto$ Gmin($D^k)$ is the left adjoint of the function $n\mapsto$ Sp$(U,n)$. This allows us to determine Gmin($D^k)$ in many cases. E.g., for a 5-element distributive lattice $D$, Gmin($D^{2023})=18$ if $D$ is a chain and Gmin($D^{2023})=15$ otherwise. It follows that large direct powers of small distributive lattices are appropriate for our 2021 cryptographic authentication protocol.