受约束图绘制的更精确拉姆齐定理
A sharper Ramsey theorem for constrained drawings
AI总结：
本文针对受约束图绘制证明了更精确的拉姆齐型定理，给出完全图受约束副本存在的顶点数上界，并由此将海莱型定理中的塔型界大幅改进为多项式界。
摘要：
给定一个图 $G$ 和一个由 $G$ 的顶点子集索引的 $\mathbb{R}^d$ 的子集族 $\mathcal C$，图 $G$ 的一个受约束绘制是指一种绘制方式，其中每条边都画在 $\mathcal C$ 中的某个集合内，并且要求不相邻的边画在索引不相交的集合中。本文证明了此类绘制的一个拉姆齐型结果。进一步地，我们展示了该结果如何用于获得海莱型定理。更精确地说，我们证明了以下结论。对每个 $n$ 和 $b$，存在 $N=O(b^{2n-3})$，具有如下性质：若 $G$ 是 $N$ 个顶点上的一个图的绘制，且 $\mathcal C$ 是 $\mathbb{R}^d$ 的一个集合族，使得每个 $(b+1)$ 元顶点组 $T$ 都位于一个以 $T$ 为索引的集合中，并且该集合包含 $T$ 中至少一条边，则在 $G$ 中我们可以找到一个受约束的完全图 $K_n$ 的副本。作为直接推论，我们得到如下海莱型结果：对每个 $d$，存在一个次数至多为 $2d+3$ 的多项式 $h(b)$，使得以下结论成立。对 $\mathbb{R}^d$ 中的任意集合族 $\mathcal F$，只要任意非空子族的交至多有 $b$ 个道路连通分支，并且平凡同调群 $H_1$，$H_2$，……，$H_{\lceil d/2\rceil-1}$，则其海莱数至多为 $h(b)$。这极大地改进了 Matoušek 的原始定理，该定理具有更强的假设并且对 $h(b)$ 给出了塔型界。在相同假设下，我们的技术也可用于给出拉东数的界。
英文摘要：
Given a graph $G$ and a collection $\mathcal C$ of subsets of $\mathbb{R}^d$ indexed by the subsets of vertices of $G$, a constrained drawing of $G$ is a drawing, where each edge is drawn inside some set from $\mathcal C$, in such a way that non-adjacent edges are drawn in sets with disjoint indices. In this paper we prove a Ramsey type result for such drawings. Furthermore we show how the result can be used to obtain Helly type theorems. More precisely, we prove the following. For each $n$ and $b$, there is $N=O(b^{2n-3})$ with the following properties: If $G$ is a drawing of a graph on $N$ vertices and $\mathcal C$ is a collection of sets of $\mathbb{R}^d$ such that each $(b+1)$-tuple $T$ of vertices lies in a set indexed by $T$ and contains at least one edge in $T$, then in $G$, we can find a constrained copy of the complete graph $K_n$. As a direct consequence we obtain the following Helly type result: For each $d$, there is a polynomial $h(b)$ of degree at most $2d+3$ such that the following holds. For every family $\mathcal F$ of sets in $\mathbb{R}^d$, its Helly number is at most $h(b)$, provided that the intersection of any non-empty subfamily has at most $b$ path-connected components, and trivial homology groups $H_1$, $H_2$, .... $H_{\lceil d/2\rceil-1}$. This dramatically improves the original theorem by Matoušek which had stronger assumption and a tower-like bound on $h(b)$. Under the same assumptions, our technique can also be used to bound Radon numbers.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
