Let $G$ be a $k$-connected graph. An $(x,U)$-fan is a set $U\subseteq V(G)$ of size $|U|\ge k$ together with a vertex $x\in V(G)\backslash U$ and a set of disjoint $(x,U)$-paths whose only common vertex is $x$. The number of disjoint $(x,U)$-paths is the size of the $(x,U)$-fan.

The problem is to show that in a $k$-connected graph there is always an $(x,U)$-fan of size $k$.

I was thinking induction over $k$, but the inductive step is rather messy.

**[Edit:]** Presumably, this question is asking to prove Dirac's fan lemma, which states that in a $k$-connected graph $G$, for every vertex $x$ and every $U \subseteq V(G)\setminus\{x\}$ with $|U| \geq k$, there is an $(x, U)$-fan of size $k$.