For $n\in\mathbb{N}$ we consider the set $\{1,\ldots,n\}$ and define the *line graph $L(K_n)$ of the complete graph $K_n$* as follows:

- $V(L(K_n)) = \big\{\{a,b\}: a,b\in \{1,\ldots, n\}, a\neq b \big\}$;
- $E(L(K_n)) = \big\{\{x,y\}: x,y\in V(L(K_n)) \text{ and } x\cap y \neq \emptyset\big\}$.

For any graph $G=(V,E)$ the *Hadwiger number* $\eta(G)$ is the largest $n\in \omega$ such that $K_n$ is a minor of $G$.

In this post it is shown that $\eta(L(K_5)) \geq 6$, and it's easy to see that $\eta(L(K_n)) \geq n+1$ for $n\geq 5$.

**Question**: Given a positive integer $k\in\mathbb{N}$, is there $n\in \mathbb{N}$ such that $\eta(L(K_n)) \geq k\cdot n$?