  We prove that every graph with $n$ vertices and at least $5n-8$ edges
contains the Petersen graph as a minor, and this bound is best possible.
Moreover we characterise all Petersen-minor-free graphs with at least $5n-11$
edges. It follows that every graph containing no Petersen minor is 9-colourable
and has vertex arboricity at most 5. These results are also best possible.
