Minimum-sized generating sets of the direct powers of the free distributive lattice on three generators and a Sperner theorem

Gábor Czédli

Published 2023-09-25Version 1

Let FD(3) denote the free distributive lattice on three generators. For each positive integer $k$, we determine the minimum size $m$ of a generating set of the $k$-th direct power FD(3)$^k$ of FD(3) up to ``accuracy $1/2$'' in the sense that we can give only the set $\{m,m+1\}$ rather than $m$. For infinitely many values of $k$, we give $m$ exactly. Furthermore, to provide a tool for determining $m$ above, we prove a Sperner (type) theorem for the 3-crown, which is the 6-element poset (partially ordered set) formed by the proper but nonempty subsets of a 3-element set. Namely, we give lower and upper bounds for the maximum number of pairwise unrelated copies of the 3-crown in the powerset lattice of an $n$-element finite set.