The extremal function for K8- 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 (11n - 35)/2 edges either has a K-8(-) minor, or is isomorphic to a graph obtained from disjoint copies of K-1,K-2,K-2,K-2,K-2 and/or K-7 by identifying cliques of size five. (c) 2005 Elsevier Inc. All rights reserved.