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Helly-type problems from a topological perspective

Pavel Paták, Zuzana Patáková

TL;DR

The paper surveys how Helly-type results extend when convexity is replaced by topological hypotheses, contrasting nerve-lemma-based proofs with non-embeddability approaches. It organizes the theory around generalized convexity, the multinerve, and homological tools such as constrained chain maps and homological minors, and shows how Ramsey-type selections yield bounds under homology constraints. The main contributions include a cohesive framework that links nerve-based and non-embeddability methods, establishes fractional and colorful extensions under $d$-Leray or related conditions, and outlines constructive schemes for Radon/Helly bounds. It also highlights open problems and recent advances with implications for combinatorial topology and geometric transversal theory.

Abstract

We discuss recent progress on topological Helly-type theorems and their variants. We provide an overview of two different proof techniques, one based on the nerve lemma, while the other on non-embeddability.

Helly-type problems from a topological perspective

TL;DR

The paper surveys how Helly-type results extend when convexity is replaced by topological hypotheses, contrasting nerve-lemma-based proofs with non-embeddability approaches. It organizes the theory around generalized convexity, the multinerve, and homological tools such as constrained chain maps and homological minors, and shows how Ramsey-type selections yield bounds under homology constraints. The main contributions include a cohesive framework that links nerve-based and non-embeddability methods, establishes fractional and colorful extensions under -Leray or related conditions, and outlines constructive schemes for Radon/Helly bounds. It also highlights open problems and recent advances with implications for combinatorial topology and geometric transversal theory.

Abstract

We discuss recent progress on topological Helly-type theorems and their variants. We provide an overview of two different proof techniques, one based on the nerve lemma, while the other on non-embeddability.
Paper Structure (9 sections, 30 theorems, 12 equations, 1 figure)

This paper contains 9 sections, 30 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Topological Helly-type theorems from the nerve lemma
  3. Topological properties of the nerve
  4. Helly type theorems from non-embeddability
  5. Generalized convexity
  6. Constrained chain maps & homological minors
  7. Ramsey
  8. Proof sketch of Proposition \ref{['p:chain_map']}.
  9. Open problems

Key Result

Theorem 1

Given a finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$, either $\bigcap \mathcal{F}\neq \emptyset$, or one can find a subfamily $\mathcal{G}\subseteq \mathcal{F}$ that has at most $d+1$ elements and satisfies $\bigcap \mathcal{G}=\emptyset$.

Figures (1)

  • Figure 1: An example of a multinerve

Theorems & Definitions (44)

  • Theorem 1: Helly's theorem h-umkkm-23
  • Definition 2: Helly number Danzer_Grunbaum_Klee_Helly_number
  • Definition 3: Nerve of a family of sets nervedef
  • Definition 4: Missing face
  • Theorem 5
  • Theorem 6
  • proof : Proof of Theorem \ref{['thm:hellyTop']} from Theorem \ref{['thm:nerveLeray']}
  • Theorem 7: Montejano-Berge13
  • Theorem 8: multinerve13
  • Definition 9: multinerve13
  • ...and 34 more