The extremal function for \(K_{8}^{-}\) minors (Q2573648)

A result of Mader from 1968, which was instrumental for the proof of the special case of Hadwiger's conjecture by Robertson, Seymour and Thomas for graphs without \(K_6\) minor, states that for \(p\leq 7\) every graph \(G\) with sufficiently many edges has a \(K_p\) minor. In the present paper the author proves a conjecture of Jakobsen from 1983 which in some sense generalizes Mader's result and might be useful to achieve progress on Hadwiger's conjecture. Specifically, he proves that every graph of order \(n\geq 8\) with at least \((11n-35)/2\) edges either has a \(K_8^-\) minor (i.e. \(K_8\) minus one edge) or is obtained from disjoint copies of \(K_{1,2,2,2,2}\) and/or \(K_7\) by identifying cliques of size five.