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. The same result holds also for a wider class of families which consist of \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：集合族$\mathcal{F}$满足$(p,q)$-性质，如果从$\mathcal{F}$中任意选取$p$个成员，总有$q$个成员能被一个单点所刺穿。Alon和Kleitman著名的$(p,q)$-定理断言：对于任意$p \geq q \geq d+1$，任何由$\mathbb{R}^d$中紧凸集组成的、满足$(p,q)$-性质的集合族$\mathcal{F}$，均可被有限个点$c(p,q,d)$所刺穿。Alon和Kalai得到了一个关于$(d-1)$维平面（称为$(d-1)$-横截）刺穿的类似定理。本文证明了以下结果，该结果可视为关于$k$-横截的一个$(\aleph_0,k+2)$-定理：设$\mathcal{F}$是$\mathbb{R}^d$中一个由闭球组成的无限族，且$0 \leq k < d$。如果从$\mathcal{F}$中任意选取$\aleph_0$个元素，总有$k+2$个元素能被一个$k$维平面所刺穿，那么$\mathcal{F}$可被有限个$k$维平面所刺穿。该结果同样适用于由下文定义的“近球体”构成的更广泛族类。这是首个将假设弱化为$(\infty,\cdot)$假设的$(p,q)$-定理。我们的证明结合了几何与拓扑工具。