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A fractional Helly theorem for set systems with slowly growing homological shatter function

Marguerite Bin

Abstract

We study parameters of the convexity spaces associated with families of sets in $\mathbb{R}^d$ where every intersection between $t$ sets of the family has its Betti numbers bounded from above by a function of $t$. Although the Radon number of such families may not be bounded, we show that these families satisfy a fractional Helly theorem. To achieve this, we introduce graded analogues of the Radon and Helly numbers. This generalizes previously known fractional Helly theorems.

A fractional Helly theorem for set systems with slowly growing homological shatter function

Abstract

We study parameters of the convexity spaces associated with families of sets in where every intersection between sets of the family has its Betti numbers bounded from above by a function of . Although the Radon number of such families may not be bounded, we show that these families satisfy a fractional Helly theorem. To achieve this, we introduce graded analogues of the Radon and Helly numbers. This generalizes previously known fractional Helly theorems.

Paper Structure

This paper contains 11 sections, 6 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Graded parameters
  3. Growth rate of the graded numbers.
  4. Toward a fractional Helly theorem.
  5. What about the homological shatter function ?
  6. Bounding the Helly number.
  7. Bounding the $\mathop{\mathrm{h}}\nolimits(\mathcal{F})$-th clique number.
  8. Definition of the function $\Psi_{d,b_0}$.
  9. Some examples
  10. Families with a given homological shatter function.
  11. Growth of the Radon numbers.

Key Result

Proposition 2.1

If $\mathop{\mathrm{h}}\nolimits^{(t)}(\mathcal{F})< t$ for all $t>t_0$, then $\mathop{\mathrm{h}}\nolimits({\mathcal{F}})\leq t_0$.

Theorems & Definitions (11)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3: Holmsen2020
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 1 more