Bollobás–Leader directed vertex-disjoint paths conjecture

Let $Q_n$ be the $n$-dimensional hypercube, identified with the power set $\mathcal{P}[n]$. For disjoint subsets $A,B\subseteq Q_n$, let $\overrightarrow{p}_v(A,B)$ denote the maximum size of a collection of directed paths between $A$ and $B$ whose interiors are pairwise vertex-disjoint. Let $BL_v(|A|,|B|)$ denote the Bollobás–Leader lower bound for the corresponding maximum number of paths. Bollobás–Leader's directed vertex-paths conjecture. If $A$ and $B$ are disjoint non-empty subsets of $Q_n$, then

$$\overrightarrow{p}_v(A,B)\geq BL_v(|A|,|B|).$$

In particular, if $|A|=|B|=\sum_{i=0}^k \binom{n}{k}$, then

$$\overrightarrow{p}_v(A,B)\geq \binom{n}{k+1}.$$

This is the directed analogue of the Bollobás–Leader vertex-paths theorem, asking whether the same bounds hold for directed paths between the relevant up-sets and down-sets. The paper proves the conjecture.