The Directed Grid Theorem, stating that a directed graphs of sufficiently large directed treewidth contains a big directed grid as a butterfly minor, after being a conjecture for nearly 20 years, has been proven in 2015 by Kawarabayashi and Kreutzer. However, the proof yields very bad dependence of ``large'' and ``big'' in the statement. In this work, we show that if one relaxes \emph{directed grid} to \emph{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 $k$.
翻译：Kawarabayashi 和 Kreutzer 2015 年, Kawarabayashi 和 Kreutzer 证明了 。 然而, 证据显示, 语句中“ 大” 和“ 大” 的依赖性非常差。 在这项工作中, 我们显示, 如果将 emph{ directed complet} 放松到 \ emph{blble of stant closy}, 就可以获得一个多面形的网格。 更确切地说, 我们显示, 每1美元中的每一美元都存在=\ mathcal{O} (k ⁇ 48}\log ⁇ 13} k), 因此, 每张方向直线图中至少有1美元包含一串的拥堵, 最多为8美元, 大小至少为1 美元。
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