Linear-sized minors with given edge density

It is proved that for every $\epsilon >0$, there exists $K>0$ such that for every integer $t\ge 2$, every graph with chromatic number at least $Kt$ contains a minor with $t$ vertices and edge density at least $1-\epsilon$. Indeed, building on recent work of Delcourt and Postle on linear Hadwiger’s conjecture, for $\epsilon \in (0,\frac{1}{256})$ we can take $K=C\log \log (1\mathrm{/}\epsilon )$ 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.