凸集并的$(p,q)$定理的新定量界

中文摘要 AI 辅助

$\boldsymbol{\text{R}}^d$中的集合若为至多$s$个凸集的并，则称为$s$-凸集。若集合族$F$中任意$p$个集合里存在$q$个相交，则称$F$满足$(p,q)$性质。设$\text{HD}_d^{(s)}(p,q)$为满足$(p,q)$性质的有限$s$-凸集族的击中集（即刺穿集）所需的最少点数。Alon与Kalai（1995）证明了对任意$p \geq q \geq d+1$及任意$s \geq 1$，$\text{HD}_d^{(s)}(p,q)$存在，但他们得到的定量界非常宽松。本文给出了一般$d$及直线上的$s$-区间（即$\text{HD}_1^{(s)}(p,q)$）的若干改进的上下界，具体结果如下：(i) 对任意$d \geq 2$、$s \geq 1$及$\boldsymbol{\text{δ}}>0$，若$p>q$且$q \geq C_d \log(e sp)$，则$\text{HD}_d^{(s)}(p,q) \leq p-q+1 + O_{d,\delta}((s \cdot \tfrac{p}{q} \cdot \log \tfrac{esp}{q})^{\rho_d+\delta})$，其中$\rho_d<d$是Rubin（2022）弱epsilon-网定理中的指数；(ii) 对任意$s \geq 1$、$p \geq q \geq 2$且$q \geq C_0 s \log(2s) \log(ep)$，有$p-q+s \leq \text{HD}_1^{(s)}(p,q) \leq p-q+2s+1$，该结果首次为$q>2$时的$\text{HD}_1^{(s)}(p,q)$提供了近紧估计；(iii) 对任意固定的$s$，存在整数$\kappa_s \in \{s, \ldots, 2s\}$及常数$C_s,p_s>0$，使得当$p \geq p_s$且$q \geq C_s \log(ep)$时，$\text{HD}_1^{(s)}(p,q) \in \{p-q+\kappa_s, \\\\;p-q+\kappa_s+1\}$，尽管阈值的精确值仍未知，但该二值集中结果成立；(iv) 对任意$s \geq 1$，$\text{HD}_3^{(s)}(p,4) \geq sp^{2-o(1)}$，该结果已显著改进了所有$d \geq 3$时凸集族（即$s=1$）上$\text{HD}_d^{(1)}(p,d+1)$的最佳已知下界。

英文摘要

A set in $\mathbb{R}^d$ is $s$-convex if it is the union of at most $s$ convex sets. A family $F$ satisfies the $(p,q)$ property if among any $p$ sets in $F$, some $q$ intersect. Let $\mathrm{HD}_d^{(s)}(p,q)$ be the minimum number of points needed to pierce a finite family of $s$-convex sets that satisfies the $(p,q)$-property. Alon and Kalai (1995) proved that $\mathrm{HD}_d^{(s)}(p,q)$ exists for any $p \geq q \geq d+1$ and any $s \geq 1$, but the quantitative bounds they obtained are very loose. We present several improved upper and lower bounds, for a general $d$ and for $s$-intervals of the line (i.e., $\mathrm{HD}_1^{(s)}(p,q)$). In particular, we prove the following: (i) For every $d\ge2$, $s \geq 1$ and $δ>0$, if $p>q$ and $q\ge C_d\log(e sp)$, then $\mathrm{HD}_d^{(s)}(p,q) \le p-q+1 + O_{d,δ}((s \cdot \tfrac{p}{q} \cdot \log \tfrac{esp}{q})^{ρ_d+δ}),$ where $ρ_d<d$ is the exponent in the weak epsilon-net theorem of Rubin (2022). (ii) For $s \geq 1$, $p \geq q \geq 2$ and $q\ge C_0s\log(2s)\log(ep)$, $p-q+s \leq \mathrm{HD}_1^{(s)}(p,q) \leq p-q+2s+1$. This result provides the first near-tight estimate for $\mathrm{HD}_1^{(s)}(p,q)$ for $q>2$. (iii) For any fixed $s$, there are an integer $κ_s\in\{s,\ldots,2s\}$ and constants $C_s,p_s>0$ such that, whenever $p\ge p_s$ and $q\ge C_s\log(ep)$, $ \mathrm{HD}_1^{(s)}(p,q)\in\{p-q+κ_s,\;p-q+κ_s+1\}. $ Interestingly, this two-value concentration result holds, although the exact value of the threshold remains unknown. (iv) For any $s \geq 1$, $\mathrm{HD}_3^{(s)}(p,4) \geq sp^{2-o(1)}$. Already for families of convex sets, this significantly improves the best known lower bound on $\mathrm{HD}_d^{(1)}(p,d+1)$, for all $d \geq 3$.

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21 pages