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

Extremal results regarding $K_6$-minors in graphs of girth at least 5