标题： 一个$(\aleph_0,k+2)$定理对于$k$穿越 显示英文标题

标题： An $(\aleph_0,k+2)$-Theorem for $k$-Transversals

摘要： 一个集合族$\mathcal{F}$满足$(p,q)$-性质，如果在$\mathcal{F}$的每一个$p$个成员中，可以被一个点穿透的某个$q$。 著名的$(p,q)$定理由 Alon 和 Kleitman 提出，断言对于任何$p \geq q \geq d+1$，任何在$\mathbb{R}^d$中满足$(p,q)$性质的紧凸集族$\mathcal{F}$可以被有限数量的点穿透$c(p,q,d)$。 关于用$(d-1)$-维平面穿刺的类似定理，称为$(d-1)$-横截面，由Alon和Kalai获得。在本文中，我们证明了以下结果，该结果可以被视为关于$(\aleph_0,k+2)$-定理的$k$-横截面：令$\mathcal{F}$为$\mathbb{R}^d$中闭球的无限族，令$0 \leq k < d$。 如果在$\mathcal{F}$的每一个$\aleph_0$个元素中，某些$k+2$可以被一个$k$维的平面穿透，那么$\mathcal{F}$可以被有限数量的$k$维平面穿透。 我们由此结果作为更一般结果的一个推论，该结果证明了对于称为\emph{近似球体}的非必须凸对象族也具有相同的断言，将在下面定义。 这是第一个$(p,q)$定理，在该定理中假设被削弱为一个$(\infty,\cdot)$假设。 我们的证明结合了几何和拓扑工具。 显示英文摘要

摘要： A family $\mathcal{F}$ of sets satisfies the $(p,q)$-property if among every $p$ members of $\mathcal{F}$, some $q$ can be pierced by a single point. The celebrated $(p,q)$-theorem of Alon and Kleitman asserts that for any $p \geq q \geq d+1$, any family $\mathcal{F}$ of compact convex sets in $\mathbb{R}^d$ that satisfies the $(p,q)$-property can be pierced by a finite number $c(p,q,d)$ of points. A similar theorem with respect to piercing by $(d-1)$-dimensional flats, called $(d-1)$-transversals, was obtained by Alon and Kalai. In this paper we prove the following result, which can be viewed as an $(\aleph_0,k+2)$-theorem with respect to $k$-transversals: Let $\mathcal{F}$ be an infinite family of closed balls in $\mathbb{R}^d$, and let $0 \leq k < d$. If among every $\aleph_0$ elements of $\mathcal{F}$, some $k+2$ can be pierced by a $k$-dimensional flat, then $\mathcal{F}$ can be pierced by a finite number of $k$-dimensional flats. We derive this result as a corollary of a more general result which proves the same assertion for families of not necessarily convex objects called \emph{near-balls}, to be defined below. This is the first $(p,q)$-theorem in which the assumption is weakened to an $(\infty,\cdot)$ assumption. Our proofs combine geometric and topological tools.