排除格点定理的紧界
Towards Tight(er) Bounds for the Excluded Grid Theorem
摘要：
我们研究了排除网格定理，图论中的一个基本结构结果，这是由罗伯逊和西摩在他们的开创性工作中证明的。该定理指出存在一个函数$f: \mathbb{Z}^+ \to \mathbb{Z}^+$，使得对于每个整数$g>0$，每个树宽至少为$f(g)$的图包含$(g\times g)$ -网格作为子图。对于每个整数$g>0$，设$f(g)$是定理成立的最小值。建立$f(g)$上的紧界是一个重要的图论问题。Robertson和Seymour证明了$f(g)=Ω(g^2\log g)$必须成立。长期以来，$f(g)$的最佳已知上界在$g$中是超指数的。Chekuri和Chuzhoy证明了$f(g)=O(g^{98}\text{poly}\log g)$的第一多项式上界.后来改进为$f(g) = O(g^{36}\text{poly} \log g)$，再改进为$f(g)=O(g^{19}\text{poly}\log g)$。本文进一步将此界改进为$f(g)=O(g^{9}\text{poly} \log g)$ .我们相信，我们的证明是显着简单比以前的界限的证明。此外，虽然有天然的障碍，似乎阻止以前的方法从产生严格的界限定理，似乎可以想象，本文提出的技术可以导致更严格的界限$f(g)$。
英文摘要：
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 $f(g)=Ω(g^2\log g)$ 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 $f(g)=O(g^{98}\text{poly}\log g)$ was proved by Chekuri and Chuzhoy. It was later improved to $f(g) = O(g^{36}\text{poly} \log g)$, and then to $f(g)=O(g^{19}\text{poly}\log g)$. In this paper we further improve this bound to $f(g)=O(g^{9}\text{poly} \log g)$. 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 the 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 $f(g)$.
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