For positive integers $m, d\in \mathbb{N}$ consider the following statement:



> 
> $\mathsf{S}(m,d):$ There is $N\in\mathbb{N}$ such that the complete graph $K\_m$ on $m$ vertices is a minor of any finite simple undirected graph $G=(V,E)$ with diameter $d$ and $|V|\ge N$.
> 
> 
> 


Does $\mathsf{S}(m,d)$ hold for any choice of positive integers $m,d$? If not, does $\mathsf{S}(m,2)$ hold for all $m\in\mathbb{N}$?



Complete minors in graphs of bounded diameter
No, because there exists trees with small diameter.


For example, the star on $N$ vertices has diameter $2$ and does not even contain a $K\_3$ minor. 


