Extremal results regarding K₆-minors in graphs of girth at least 5

We prove that every 6-connected graph of girth $\geq 6$ has a $K_6$-minor and thus settle the Jorgensen conjecture for graphs of girth $ \geq 6$. Relaxing the assumption on the girth, we prove that every 6-connected $n$-vertex graph of size $\geq 3 1/5 n-8$ and of girth $\geq 5$ contains a $K_6$-minor.