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 bound known for an arbitrary planar H is O(|V(H)|⁹poly~log|V(H)|). We show that if H is the disjoint union of cycles, then $f(H)$ is O(|V(H)|log² |V(H)|), which is a log|V(H)| factor away from being optimal.