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.

翻译：若集合族$\mathcal{F}$满足：对其任意$p$个成员，其中总有$q$个能被单个点刺穿，则称该族满足$(p,q)$性质。Alon与Kleitman著名的$(p,q)$定理断言：对任意满足$p \geq q \geq d+1$的整数及$\mathbb{R}^d$中满足$(p,q)$性质的紧凸集族$\mathcal{F}$，总存在有限个点$c(p,q,d)$刺穿该族。关于$(d-1)$维平面（称为$(d-1)$-横截）刺穿的类似定理由Alon与Kalai获得。本文证明以下结果，可视为关于$k$-横截的$(\aleph_0,k+2)$定理：设$\mathcal{F}$为$\mathbb{R}^d$中闭球构成的无限族，且$0 \leq k < d$。若对$\mathcal{F}$中任意$\aleph_0$个元素，其中总有$k+2$个能被某个$k$维平面刺穿，则$\mathcal{F}$可被有限个$k$维平面刺穿。我们将此结果作为更一般定理的推论导出，该定理对未必凸的\emph{近球}对象族（定义见下文）证明了相同结论。这是首个将假设弱化为$(\infty,\cdot)$型条件的$(p,q)$定理。我们的证明结合了几何与拓扑工具。