Bounds on treewidth via excluding disjoint unions of cycles

One of the fundamental results in graph minor theory is that for every planar graph~$H$, there is a minimum integer~$f(H)$ such that graphs with no minor isomorphic to~$H$ have treewidth at most~$f(H)$. The best known bound for an arbitrary planar $H$ is $O(\mathrm{\mid }V(H){\mathrm{\mid }}^9\mathrm{poly log}\mathrm{\mid }V(H)\mathrm{\mid })$. We show that if $H$ is the disjoint union of cycles, then $f(H)$ is $O(\mathrm{\mid }V(H)\mathrm{\mid }{\log }^2\mathrm{\mid }V(H)\mathrm{\mid })$, which is a $\log \mathrm{\mid }V(H)\mathrm{\mid }$ factor away being optimal.