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

Chaya Keller, Micha A. Perles

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. 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.

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

Abstract

A family of sets satisfies the -property if among every members of , some can be pierced by a single point. The celebrated -theorem of Alon and Kleitman asserts that for any , any family of compact convex sets in that satisfies the -property can be pierced by a finite number of points. A similar theorem with respect to piercing by -dimensional flats, called -transversals, was obtained by Alon and Kalai. In this paper we prove the following result, which can be viewed as an -theorem with respect to -transversals: Let be an infinite family of closed balls in , and let . If among every elements of , some can be pierced by a -dimensional flat, then can be pierced by a finite number of -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 -theorem in which the assumption is weakened to an assumption. Our proofs combine geometric and topological tools.
Paper Structure (9 sections, 7 theorems, 16 equations, 6 figures)

This paper contains 9 sections, 7 theorems, 16 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Our contributions
  4. Follow-up work
  5. Definitions, Notations, and Basic Observations
  6. Proof of the Main Theorem
  7. The induction basis for $k=0$ and every $d$
  8. The induction step
  9. Proof of Proposition \ref{['thm:ex']}

Key Result

Theorem 1.1

For any triple of positive integers $p \geq q \geq d+1$ there exists $c=c(p,q,d)$ such that if $\mathcal{F}$ is a family of compact convex sets in $\mathbb{R}^d$ that satisfies the $(p,q)$-property, then there exists a transversal for $\mathcal{F}$ of size at most $c$.

Figures (6)

  • Figure 1: An illustration for the proof of Case 1 for $k=1, m=d=2$.
  • Figure 2: An illustration for the proof of Claim \ref{['cl:cone']}.
  • Figure 3: An illustration for the proof of Claim \ref{['cl:wide-cone']}.
  • Figure 4: An illustration for the inductive construction in Case 2, where $k=1$ and $d=2$.
  • Figure 5: An illustration for Claim \ref{['cl:KtoK']}.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 1.1: the $(p,q)$-theorem AK
  • Theorem 1.2: AK95
  • Definition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 2.1
  • Lemma 3.1
  • Claim 3.2
  • Remark 3.3
  • ...and 5 more