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Colorful Helly via induced matchings

Cosmin Pohoata, Kevin Yang, Shengtong Zhang

TL;DR

This work analyzes colorful Helly theorems through the lens of induced matchings in the bipartite complement of the incidence graph between a ground set X and a set family ℱ, proving the central bound η(X,ℱ) ≤ 1 + τ(X,ℱ) and introducing the refined bound via τ′(X,ℱ). When τ′ = τ − 1, the bound tightens to η(X,ℱ) = τ(X,ℱ), enabling optimal colorful Helly results in natural geometric settings: for d-dimensional spheres, η = d+2; for hypersurfaces of bounded degree, η ≤ binom{D+d}{d}; and for Hamming balls of radius t, η = 2^{t+1}. The paper provides sharpness examples, Noetherian considerations removing finiteness hypotheses, and fractional Helly consequences, linking combinatorial incidence structures to Helly-type outcomes. It also connects these combinatorial bounds to topological properties of nerve complexes, showing that bounded comatching does not generally force collapsibility or Leray properties, and explores the limits via joins and higher-dimensional constructions.

Abstract

We establish a theorem regarding the maximum size of an {\it{induced}} matching in the bipartite complement of the incidence graph of a set system $(X,\mathcal{F})$. We show that this quantity plus one provides an upper bound on the colorful Helly number of this set system, i.e. the minimum positive integer $N$ for which the following statement holds: if finite subfamilies $\mathcal{F}_1,\ldots, \mathcal{F}_{N} \subset \mathcal{F}$ are such that $\cap_{F \in \mathcal{F}_{i}} F = 0$ for every $i=1,\ldots,N$, then there exists $F_i \in \mathcal{F}_i$ such that $F_1 \cap \ldots \cap F_{N} = \emptyset$. We will also discuss some natural refinements of this result and applications.

Colorful Helly via induced matchings

TL;DR

This work analyzes colorful Helly theorems through the lens of induced matchings in the bipartite complement of the incidence graph between a ground set X and a set family ℱ, proving the central bound η(X,ℱ) ≤ 1 + τ(X,ℱ) and introducing the refined bound via τ′(X,ℱ). When τ′ = τ − 1, the bound tightens to η(X,ℱ) = τ(X,ℱ), enabling optimal colorful Helly results in natural geometric settings: for d-dimensional spheres, η = d+2; for hypersurfaces of bounded degree, η ≤ binom{D+d}{d}; and for Hamming balls of radius t, η = 2^{t+1}. The paper provides sharpness examples, Noetherian considerations removing finiteness hypotheses, and fractional Helly consequences, linking combinatorial incidence structures to Helly-type outcomes. It also connects these combinatorial bounds to topological properties of nerve complexes, showing that bounded comatching does not generally force collapsibility or Leray properties, and explores the limits via joins and higher-dimensional constructions.

Abstract

We establish a theorem regarding the maximum size of an {\it{induced}} matching in the bipartite complement of the incidence graph of a set system . We show that this quantity plus one provides an upper bound on the colorful Helly number of this set system, i.e. the minimum positive integer for which the following statement holds: if finite subfamilies are such that for every , then there exists such that . We will also discuss some natural refinements of this result and applications.

Paper Structure

This paper contains 10 sections, 17 theorems, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Warm up: Colorful Helly for hypersurfaces
  3. Spheres
  4. Hypersurfaces
  5. Proof of Theorem \ref{['main']}, Proposition \ref{['noetherian']} and Theorem \ref{["main'"]}
  6. Proof of Theorem \ref{["main'"]}
  7. Examples showing sharpness
  8. Proof of Proposition \ref{['noetherian']}
  9. Applications
  10. Collapsibility

Key Result

Theorem 1.1

Let $X$ be a set and let $\mathscr{F}$ be a collection of subsets of $X$. Then we have

Figures (2)

  • Figure 1: Four spheres in $\mathbb{R}^2$ forming a comatching.
  • Figure 2: The regions in red and blue are members of $\mathscr{F}$. To visualize $K = N(\mathscr{F})$, we identify the red region with grid square $1$, and the blue region with grid square $13$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 2.1: Colorful Helly for spheres
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3: Colorful Helly for complex polynomials
  • Lemma 2.4
  • ...and 21 more