Boolean lattice ($\mathcal{P}([n])$)

The Boolean lattice is the power set of a finite set ordered by inclusion. Its rank-$h$ level consists of all $h$-element subsets.

An up-set is a family $\mathcal{U}$ such that $A\in \mathcal{U}$ and $A\subseteq B$ imply $B\in \mathcal{U}$.

A down-set is a family $\mathcal{D}$ such that $A\in \mathcal{D}$ and $B\subseteq A$ imply $B\in \mathcal{D}$. It is the complement of an up-set within the same Boolean lattice.

An antichain is a family of sets no one of which contains another.

A uniformly random maximal chain is obtained from a uniformly random ordering of the ground set by taking its successive initial segments. A fixed $h$-element set occurs with probability ${\left(\genfrac{}{}{0ex}{}{n}{h}\right)}^{-1}$.

If $\mathcal{A}$ is an antichain in $\mathcal{P}([n])$ and ${\mathcal{A}}_h$ is its rank-$h$ part, thenThe left side is the expected number of members of $\mathcal{A}$ on a Uniformly random maximal chain in a Boolean lattice.

For $\mathcal{A}\subseteq [n{]}^{(r)}$, the local LYM inequality saysIt follows by counting pairs $(B,A)$ with $B\subset A$, $|B|=r-1$, and $A\in \mathcal{A}$.

Every antichain in $\mathcal{P}([n])$ has at most $\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$ members. Equality is attained only by a full middle level, with either middle level possible when $n$ is odd.

If set families $\mathcal{A},\mathcal{B}\subseteq \mathcal{P}([n])$ are cross-incomparable, meaning no member of either contains a member of the other, thenThe proof applies the Harris-Kleitman inequality to the upward closures generated by the two families.