Constant Congestion Brambles in Directed Graphs
Tomáš Masařík, Marcin Pilipczuk, Paweł Rzążewski, Manuel Sorge · SIAM Journal on Discrete Mathematics · 2022
The Directed Grid Theorem, stating that there is a function $f$ such that a directed graph of directed treewidth at least $f(k)$ contains a directed grid of size at least $k$ as a butterfly minor, after being a conjecture for nearly 20 years, was proved in 2015 by Kawarabayashi and Kreutzer. However, the function $f$ obtained in the proof is very fast growing. In this work, we show that if one relaxes directed grid to bramble of constant congestion, one can obtain a polynomial bound. More precisely, we show that for every $k \geq 1$ there exists $t = \mathcal{O}(k^{48} \log^{13} k)$ such that every directed graph of directed treewidth at least $t$ contains a bramble of congestion at most 8 and size at least $k$.