The extremal function for minors

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Let K 8 - be the graph obtained from K 8 by deleting one edge. We prove a conjecture of Jakobsen that every simple graph on n ⩾ 8 vertices and at least ( 11 n - 35 ) / 2 edges either has a K 8 - minor, or is isomorphic to a graph obtained from disjoint copies of K 1 , 2 , 2 , 2 , 2 and/or K 7 by identifying cliques of size five.