OPEN
Let $G$ be a graph with $1+nm$ vertices and $1+n\binom{m}{2}$ edges. Must $G$ contain two points which are connected by $m$ disjoint paths?
A conjecture of Bollobás and Erdős
[BoEr62]. This would be best possible, as demonstrated by $n$ disjoint copies of $K_m$ and a disjoint vertex.
Bollobás proved this when $m=4$ - in fact he showed that every graph with $n$ vertices and $2n-1$ edges contains two points joined by $4$ edge-disjoint paths.
