The proofs of two directed paths conjectures of Bollob\'as and Leader

Let $A$ and $B$ be disjoint sets, of size $2^k$, of vertices of $Q_n$, the $n$-dimensional hypercube. In 1997, Bollob\'as and Leader proved that there must be $(n-k)2^k$ edge-disjoint paths between such $A$ and $B$. They conjectured that when $A$ is a down-set and $B$ is an up-set, these paths may be chosen to be directed (that is, the vertices in the path form a chain). We use a novel type of compression argument to prove stronger versions of these conjectures, namely that the largest number of edge-disjoint paths between a down-set $A$ and an up-set $B$ is the same as the largest number of directed edge-disjoint paths between $A$ and $B$. Bollob\'as and Leader made an analogous conjecture for vertex-disjoint paths and we prove a strengthening of this by similar methods. We also prove similar results for all other sizes of $A$ and $B$.