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Set Systems with Covering Properties and Low VC-Dimension

George Peterzil, Johanna Steinmeyer

Abstract

Given natural numbers $k \leq s \leq n$, we ask: what is the minimal VC-dimension of a family $\mathcal{F}$ of $s$-subsets of $[n]$ that covers all $k$-subsets of $[n]$? We first show that for sufficiently large $n$ this number is always $k$, and construct families which give a lower bound for the actual growth of this stabilization point.

Set Systems with Covering Properties and Low VC-Dimension

Abstract

Given natural numbers , we ask: what is the minimal VC-dimension of a family of -subsets of that covers all -subsets of ? We first show that for sufficiently large this number is always , and construct families which give a lower bound for the actual growth of this stabilization point.
Paper Structure (3 sections, 11 theorems, 6 equations, 2 figures)

This paper contains 3 sections, 11 theorems, 6 equations, 2 figures.

Table of Contents

  1. Definitions and Examples
  2. General Bounds and Constructions
  3. Specific Families

Key Result

Theorem 1

Every family of finite sets which covers every finite subset of the first uncountable cardinal has VC-dimension at least $2$, and there is such a family with VC-dimension exactly $2$.

Figures (2)

  • Figure 1: A member of $\mathcal{F}$ for $k=5$.
  • Figure 2: $\mathcal{F}_2$ when starting with $m=4$.

Theorems & Definitions (21)

  • Theorem 1: Low, Proposition 3.3 and Theorem 3.8
  • Theorem 2: Sau,She
  • Theorem 3: \ref{['main']}
  • Definition 1.1
  • Definition 1.3
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • ...and 11 more