A graph $G$ is $m$-joined if there is an edge between every two disjoint $m$-sets of vertices. In this paper, we prove that for any $\varepsilon>0$ and sufficiently large $m, n\in \mathbb{ N}$ with $m \le n^{ 1-\varepsilon}$, every $n$-vertex $m$-joined graph $G$ contains a minor with density $\Omega\!\left(\tfrac{ n}{ \sqrt{ m}}\right)$, which is best possible up to a constant factor. When $m \ge n^{ 1-\varepsilon}$, we further show that $G$ contains a clique minor of order $\Omega\!\left(\tfrac{ n}{ \sqrt{ m\log m}}\right)$.
Categories: math.CO