Set Systems with Covering Properties and Low VC-Dimension

George Peterzil; Johanna Steinmeyer

Set Systems with Covering Properties and Low VC-Dimension

Combinatorics 2024-01-24 v1

Authors: 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.

Keywords

set theory and dimension theory combinatorial bounds and extremal problems intersecting families

Cite

@article{arxiv.2401.12252,
  title  = {Set Systems with Covering Properties and Low VC-Dimension},
  author = {George Peterzil and Johanna Steinmeyer},
  journal= {arXiv preprint arXiv:2401.12252},
  year   = {2024}
}

Related papers

View all related →