Dense Matchings of Linear Size in Graphs with Independence Number 2

For a real number $c>4$, we prove that every graph $G$ with $\alpha (G)\le 2$ and $\mathrm{\mid }V(G)\mathrm{\mid }\ge ct$ has a matching $M$ with $\mathrm{\mid }M\mathrm{\mid }=t$ such that the number of non-adjacent pairs of edges in $M$ is at most: \begin{equation*} \left( \frac{1}{c\left(c-1\right)^2} + O_c\left(t^{-1/3} \right) \right) \binom{t}{2}. \end{equation*} This is related to an open problem of Seymour (2016) about Hadwiger’s Conjecture, who asked if there is a constant $\epsilon >0$ such that every graph $G$ with $\alpha (G)\le 2$ has $\text{had}(G)\ge (\frac{1}{3}+\epsilon )\mathrm{\mid }V(G)\mathrm{\mid }$.