Linear-product bound for ℓ(p,q)

Prove the conjectured bound ℓ(p,q) = O*(pq) for the minimum treewidth threshold guaranteeing pairwise disjoint connected subgraphs of treewidth at least p whose contraction produces a minor of treewidth at least q.

Background

The conjecture concerns the parameter ℓ(p,q) introduced in the preceding open problem. The paper currently obtains only the much weaker bound O*((pq)9), via the grid-minor theorem.

The proposed O*(pq) estimate would match the natural product of the two target treewidth scales and would imply, while strengthening, Conjecture 1 of Chuzhoy and collaborators, which addresses only the existence of many vertex-disjoint subgraphs of treewidth p and does not require the additional contraction property.

References

We propose the following. \begin{conj}[$\cref{con1}$] $\ell(p,q) O\ast(pq)$. \end{conj}

— Polynomial Bounds in the Apex Minor Theorem  (2503.04228 - Hendrey et al., 6 Mar 2025) in Conjecture 1, Section 5, “Open Problems”