State and prove Menger's theorem (vertex form).

Let $G$ be a graph of connectivity $\kappa(G) \geq k$ and let $S, T$ be disjoint subsets of $V(G)$ with $|S|,|T| \geq k$. Show that there exist $k$ vertex disjoint paths from $S$ to $T$.

The graph $H$ is said to be $k$-linked if, for every sequence $s_{1}, \ldots, s_{k}, t_{1}, \ldots, t_{k}$ of $2 k$ distinct vertices, there exist $s_{i}-t_{i}$ paths, $1 \leq i \leq k$, that are vertex disjoint. By removing an edge from $K_{2 k}$, or otherwise, show that, for $k \geqslant 2$, $H$ need not be $k$-linked even if $\kappa(H) \geq 2 k-2$.

Prove that if $|H|=n$ and $\delta(H) \geq \frac{1}{2}(n+3 k)-2$ then $H$ is $k$-linked.