  It is proved that for every $\varepsilon>0$, there exists $K>0$ such that for
every integer $t\ge2$, every graph with chromatic number at least $Kt$ contains
a minor with $t$ vertices and edge density at least $1-\varepsilon$. Indeed,
building on recent work of Delcourt and Postle on linear Hadwiger's conjecture,
for $\varepsilon\in(0,\frac{1}{256})$ we can take $K=C\log\log(1/\varepsilon)$
where $C>0$ is a universal constant, which extends their recent $O(t\log\log
t)$ bound on the chromatic number of graphs with no $K_t$ minor.
