Sperner property

Remarks

Since each $P_i$ is an antichain, the condition says that the width of $P$ equals $\max_i \abs{P_i}$.

Example: the lattice of subsets of $\set{1, \ldots, n}$, ranked by size, has the Sperner property. Check: by Sperner's theorem, an antichain has at most $\binom{n}{\floor{n/2}}$ members, which is the size of the middle level.

Non-example: let $P$ have elements $a_1, a_2, b$ of rank $0$ and $c, d_1, d_2$ of rank $1$, with the relations $a_1 < c$, $a_2 < c$, $b < d_1$, $b < d_2$ and no others. Each level has $3$ elements, but $\set{a_1, a_2, d_1, d_2}$ is an antichain with $4$ elements.

A poset is strongly Sperner when, for every $k$, the largest union of $k$ antichains is no larger than the largest union of $k$ rank levels.