🤖 AI Summary
This paper addresses the $k$-dimensional plane transversal problem for infinite families of closed balls in $mathbb{R}^d$: if every countable subfamily admits a $k$-flat intersecting some $k+2$ members, does the entire family admit a finite $k$-transversal?
Method: Leveraging compactness arguments, high-dimensional combinatorial geometry, and topological analysis, the authors extend the classical $(p,q)$-theorem to the infinite setting with $p = aleph_0$, establishing the first $(aleph_0, k+2)$-type Helly theorem for $k$-transversals.
Contribution/Results: They prove that the stated condition guarantees finite $k$-transversality, thereby obtaining the first infinite $(p,q)$-type theorem for $k$-flats. The result is further generalized to near-ball families. This work establishes the first existence paradigm for transversals to countably infinite families in discrete and computational geometry, extending beyond the traditional finite $(p,q)$-theorem framework and resolving a fundamental question on infinitary Helly-type phenomena.
📝 Abstract
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.
