Let $G$ be a graph and $A, B \subset V(G)$. Show that if every $A B$-separator in $G$ has order at least $k$ then there exist $k$ vertex-disjoint $A B$-paths in $G$.

Let $k \geqslant 3$ and assume that $G$ is $k$-connected. Show that $G$ must contain a cycle of length at least $k$.

Assume further that $|G| \geqslant 2 k$. Must $G$ contain a cycle of length at least $2 k ?$ Justify your answer.

What is the largest integer $n$ such that any 3-connected graph $G$ with $|G| \geqslant n$ must contain a cycle of length at least $n$ ?

[No form of Menger's theorem or of the max-flow-min-cut theorem may be assumed without proof.]