A graph $H$ is a minor of a graph $G$ if $H$ can be obtained from $G$ by a sequence of contractions and deletions of edges. We say that $G$ has an $H$-minor if $G$ has $H$ as a minor. The complete graph on $n$ vertices is denoted by $K_n$.
The following theorem is from what I read in the article: "Suzuki, Yusuke $K_7$-minors in optimal 1-planar graphs. Discrete Math. 340, No. 6, 1227-1234 (2017)".
Theorem[Mader] A graph with $n$ vertices and at least $4n − 9$ edges has a $K_6$-minor.
- W. Mader, Homomorphiesätze für Graphen, Math. Ann. 178 (1968) 154–168.
I would like to see the proof of the theorem, but the original article is in German. I don't know where else I can see a proof (in English) of this theorem. Or it's easy to prove.
The proof of this theorem can be found in [1] (Theorem 4) by using induction on the number of vertices. The proof is quite different from the one given by Mader. That is why they include it in Section 2 of this paper.
[1]: J. Campbell, T. W. Mattman, R. Ottman, J. Pyzer, M. Rodrigues, and S. Williams. Intrinsic knotting and linking of almost complete graphs, 2007. URL https://arxiv.org/pdf/math/0701422.pdf