Cube Height, Cube Width and Related Extremal Problems for Posets

Given a poset $P$, a family $\mathcal{S}=\{S_x:x\in P\}$ of sets indexed by the elements of $P$ is called an inclusion representation of $P$ if $x\leqslant y$ in $P$ if and only if $S_x\subseteq S_y$. The cube height of a poset is the least non-negative integer $h$ such that $P$ has an inclusion representation for which every set has size at most $h$. In turn, the cube width of $P$ is the least non-negative integer $w$ for which there is an inclusion representation $\mathcal{S}$ of $P$ such that $|\bigcup\mathcal{S}|=w$ and every set in $\mathcal{S}$ has size at most the cube height of $P$. In this paper, we show that the cube width of a poset never exceeds the size of its ground set, and we characterize those posets for which this inequality is tight. Our research prompted us to investigate related extremal problems for posets and inclusion representations. Accordingly, the results for cube width are obtained as extensions of more comprehensive results that we believe to be of independent interest.