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The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\mathcal{F}$ is a family of subsets of $X$ such that no two sets $A, B \in \mathcal{F}$ satisfy $A \subset B$ (or $B \subset A$) and $A \cap X_i=B\cap X_i$ for some $i$, then $|\mathcal{F}| \le {n \choose \lfloor n/2\rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following new result which may be of independent interest: let $\mathcal{F},\mathcal{G}$ be intersecting families of subsets of an $n$-element set that are additionally cross-Sperner, meaning that if $A \in\mathcal{F}$ and $B \in \mathcal{G}$, then $A \not\subset B$ and $B \not\subset A$. Then $|\mathcal{F}| +|\mathcal{G}| \le 2^{n-1}$ and there are exponentially many examples showing that this bound is tight.
12Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets
Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets
SPERNER THEOREMS FOR UNRELATED COPIES OF POSETS AND GENERATING DISTRIBUTIVE LATTICES
For a finite poset (partially ordered set) \(U\) and a natural number \(n\), let \(S(U,n)\) 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 \(S(U,n)\) 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.J. Stahl, and W.T.Jr. Trotter. We determine \(S(U,n)\) for all finite posets with 0 and 1, and we give reasonable estimates for the "\(V\)-shaped" 3-element poset and, mainly, for the 4-element poset with 0 and three maximal elements. For a lattice \(L\), let \(G_{\min}L\) 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 \(k\mapsto G_{\min}{D^k}\) is the left adjoint of the function \(n\mapsto S(U,n)\). This allows us to determine \(G_{\min}{D^k}\) in many cases. E.g., for a 5-element distributive lattice \(D\), \(G_{\min}{D^{2023}}=18\) if \(D\) is a chain and \(G_{\min}{D^{2023}}=15\) otherwise. The present paper, another recent paper, and a 2021 one indicate that large direct powers of small distributive lattices could be of interest in cryptography.
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