弦图中的一个锐阈值现象
A sharp threshold phenomenon in string graphs
摘要：
我们证明，对于每个$ε>0$存在$δ>0$，使得以下成立。令$\mathcal{C}$是平面中的$n$曲线的集合，使得在$\mathcal{C}$中至多存在具有非空交点的曲线$\{α,β\}$的$(\frac{1}{4}-ε)\frac{n^{2}}{2}$对。则$\mathcal{C}$包含两个不相交子集$\mathcal{A}$和$\mathcal{B}$，使得$|\mathcal{A}|=|\mathcal{B}|\geq δn$和每个$α\in \mathcal{A}$与每个$β\in\mathcal{B}$不相交。另一方面，对于每个正整数$n$，存在平面中的$n$曲线的集合$\mathcal{C}$，使得至多有$(\frac{1}{4}+ε)\frac{n^{2}}{2}$对具有非空交点的曲线$\{α,β\}$，但是如果$\mathcal{A},\mathcal{B}\subset \mathcal{C}$使得对于每个$(α,β)\in \mathcal{A}\times\mathcal{B}$，$|\mathcal{A}|=|\mathcal{B}|$和$α\cap β=\emptyset$，则$|\mathcal{A}|=|\mathcal{B}|=O(\frac{1}ε\log n)$。
英文摘要：
We prove that for every $ε>0$ there exists $δ>0$ such that the following holds. Let $\mathcal{C}$ be a collection of $n$ curves in the plane such that there are at most $(\frac{1}{4}-ε)\frac{n^{2}}{2}$ pairs of curves $\{α,β\}$ in $\mathcal{C}$ having a nonempty intersection. Then $\mathcal{C}$ contains two disjoint subsets $\mathcal{A}$ and $\mathcal{B}$ such that $|\mathcal{A}|=|\mathcal{B}|\geq δn$, and every $α\in \mathcal{A}$ is disjoint from every $β\in\mathcal{B}$. On the other hand, for every positive integer $n$ there exists a collection $\mathcal{C}$ of $n$ curves in the plane such that there at most $(\frac{1}{4}+ε)\frac{n^{2}}{2}$ pairs of curves $\{α,β\}$ having a nonempty intersection, but if $\mathcal{A},\mathcal{B}\subset \mathcal{C}$ are such that $|\mathcal{A}|=|\mathcal{B}|$ and $α\cap β=\emptyset$ for every $(α,β)\in \mathcal{A}\times\mathcal{B}$, then $|\mathcal{A}|=|\mathcal{B}|=O(\frac{1}ε\log n)$.
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