Helly-type Theorems for Hollow Axis-aligned Boxes

盒子的单调性质的Helly型定理

Helly-type theorems for monotone properties of boxes

提出统一方法证明盒子单调性质的Helly型定理，涵盖大体积或含点等性质，推广到H-凸集，并给出早期结果的新证明。

Discrete quantitative Helly-type theorems with boxes

Research on Helly-type theorems in combinatorial convex geometry has produced volumetric versions of Helly's theorem using witness sets and quantitative extensions of Doignon's theorem. This paper combines these philosophies and presents quantitative Helly-type theorems for the integer lattice with axis-parallel boxes as witness sets. Our main result shows that, while quantitative Helly numbers for the integer lattice grow polynomially in each fixed dimension, their variants with boxes as witness sets are uniformly bounded. We prove several colorful and fractional variations on this theorem. W…

离散 Helly 定理在盒子上的扩展

Extensions of discrete Helly theorems for boxes

本文证明了 $\mathbb{R}^d$ 中轴平行盒子 Halman 离散 Helly 定理的彩色、分数和定量扩展版本，并将结果推广至 $H$-凸集。

盒族中成对情形的分数离散 Helly 定理

Fractional discrete Helly for pairs in a family of boxes

本文研究轴平行盒族的分数离散 Helly 性质，证明只需 $S$-相交对密度足够大，即可保证存在与族大小呈线性关系的 $S$-相交子族。

Extremal results on intersection graphs of boxes in $R^d$

The main purpose of this paper is to study extremal results on the intersection graphs of boxes in $\R^d$. We calculate exactly the maximal number of intersecting pairs in a family $\F$ of $n$ boxes in $\R^d$ with the property that no $k+1$ boxes in $\F$ have a point in common. This allows us to improve the known bounds for the fractional Helly theorem for boxes. We also use the Fox-Gromov-Lafforgue-Naor-Pach results to derive a fractional Erdős-Stone theorem for semi-algebraic graphs in order to obtain a second proof of the fractional Helly theorem for boxes.

A fractional Helly theorem for boxes

Let $\mathcal{F}$ be a family of $n$ axis-parallel boxes in $\mathbb{R}^d$ and $α\in (1-1/d,1]$ a real number. There exists a real number $β(α)>0$ such that if there are $α{n\choose 2}$ intersecting pairs in $\mathcal{F}$, then $\mathcal{F}$ contains an intersecting subfamily of size $βn$. A simple example shows that the above statement is best possible in the sense that if $α\leq 1-1/d$, then there may be no point in $\mathbb{R}^d$ that belongs to more than $d$ elements of $\mathcal{F}$.