We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function $f:{\mathbb{Z}}^{+}\to {\mathbb{Z}}^{+}$, such that for every integer $g>0$, every graph of treewidth at least $f(g)$ contains the $(g\times g)$-grid as a minor. For every integer $g>0$, let $f(g)$ be the smallest value for which the theorem holds. Establishing tight bounds on $f(g)$ is an important graph-theoretic question. Robertson and Seymour showed that $f(g)=\mathrm{\Omega }(g^2\log g)$ must hold. For a long time, the best known upper bounds on $f(g)$ were super-exponential in g. The first polynomial upper bound of $f(g)=O(g^{98}\mathrm{poly}\log g)$ was proved by Chekuri and Chuzhoy. It was later improved to $f(g)=O(g^{36}\mathrm{poly}\log g)$, and then to $f(g)=O(g^{19}\mathrm{poly}\log g)$. In this paper we further improve this bound to $f(g)=O(g^9\mathrm{poly}\log g)$. We believe that our proof is significantly simpler than the proofs of the previous bounds. Moreover, while there are natural barriers that seem to prevent previous methods from yielding tight bounds for the theorem, it seems conceivable that the techniques proposed in this paper can lead to even tighter bounds on $f(g)$.

我们研究了排除网格定理，这是图论的基本结构结果，由罗伯逊和西摩在图未成年人的开创性工作中得到证明。定理指出有一个函数$F：{\mathbb{ž}}^{+}\to {\mathbb{ž}}^{+}$，这样每个整数 $G>0$，每个树宽图至少 $F（G）$ 包含 $（G\times G）$-作为未成年人的网格。对于每个整数$G>0$，让 $F（G）$是定理成立的最小值。建立严格的界限$F（G）$是一个重要的图论问题。罗伯逊和西摩表明$F（G）=\mathrm{\Omega }（G^2\mathrm{日志}G）$必须持有。长期以来，最著名的上限$F（G）$在g中是超指数的。的第一个多项式上限$F（G）=Ø（G^{98}\mathrm{聚}\mathrm{日志}G）$Chekuri和Chuzhoy证明了这一点。后来改进为$F（G）=Ø（G^{36}\mathrm{聚}\mathrm{日志}G）$，然后 $F（G）=Ø（G^{19}\mathrm{聚}\mathrm{日志}G）$。在本文中，我们进一步改进了$F（G）=Ø（G^9\mathrm{聚}\mathrm{日志}G）$。我们认为，我们的证明比以前的界限要简单得多。而且，尽管存在自然障碍似乎阻止了先前的方法为定理产生严格的界限，但可以想象的是，本文提出的技术可以导致更严格的界限。$F（G）$。