Polynomial or single-exponential bound for congestion-one brambles in semicomplete digraphs

Determine whether the doubly exponential directed pathwidth bound for obtaining a bramble of size k and congestion 1 in semicomplete digraphs can be reduced to a polynomial or even a single-exponential bound.

Background

For semicomplete digraphs, prior work yields a doubly exponential bound through the construction of k-triples and a Ramsey-type argument. The paper identifies the dependence caused by finding large transitive tournaments in localized structures as the source of this bound and leaves open whether substantially smaller dependence is possible when the target is directly a bramble of size k and congestion 1.

References

It is not known if this bound can be made polynomial, or even reduced to a single exponential, if we request a bramble of size $k$ and congestion~$1$ instead of a $k$-triple.

— Polynomial directed treewidth forces a large bramble with congestion two  (2610.13270 - Campos et al., 9 Oct 2026) in Section 1, paragraph “Further research”