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Helly-type theorems for separated $d$-intervals

Wei Rao

TL;DR

The paper extends Helly-type theory to convexity spaces built from separated $d$-intervals by proving tight bounds: the Radon number satisfies $r(P,\, ext{C}_{ eq}(P)) \\le \,2d+1$, the Helly number satisfies $h(P,\, ext{C}_{ eq}(P)) \\le \,2d$, and the fractional Helly number equals $2$, with corresponding colorful and $(p,q)$-type theorems. The approach hinges on $d$-collapsibility and a refined analysis of intersections via a function on $d$-interval components, together with a series of collapsibility-based lemmas and probabilistic/LP techniques. The results generalize classical theorems to the discrete setting of separated $d$-intervals and establish a bridge to axis-parallel boxes, offering a toolkit for discrete convexity in combinatorial geometry. These findings have implications for transversal theory, colorful variants, and k-intersection generalizations in specialized convexity spaces.

Abstract

A separated $d$-interval is defined as a disjoint union of $d$ convex sets from the real line $\mathbb R$. In this paper, we establish a series of Helly-type theorems for convexity spaces derived from separated $d$-intervals. Our results encompass the Radon number, Helly number, colorful Helly number, fractional Helly number, colorful fractional Helly theorem, $(p,q)$ theorem, and two kinds of colorful $(p,q)$ theorems for these convexity spaces. The primary tools employed in our proofs involve simplicial complexes and collapsibility.

Helly-type theorems for separated $d$-intervals

TL;DR

The paper extends Helly-type theory to convexity spaces built from separated -intervals by proving tight bounds: the Radon number satisfies , the Helly number satisfies , and the fractional Helly number equals , with corresponding colorful and -type theorems. The approach hinges on -collapsibility and a refined analysis of intersections via a function on -interval components, together with a series of collapsibility-based lemmas and probabilistic/LP techniques. The results generalize classical theorems to the discrete setting of separated -intervals and establish a bridge to axis-parallel boxes, offering a toolkit for discrete convexity in combinatorial geometry. These findings have implications for transversal theory, colorful variants, and k-intersection generalizations in specialized convexity spaces.

Abstract

A separated -interval is defined as a disjoint union of convex sets from the real line . In this paper, we establish a series of Helly-type theorems for convexity spaces derived from separated -intervals. Our results encompass the Radon number, Helly number, colorful Helly number, fractional Helly number, colorful fractional Helly theorem, theorem, and two kinds of colorful theorems for these convexity spaces. The primary tools employed in our proofs involve simplicial complexes and collapsibility.
Paper Structure (20 sections, 18 theorems, 16 equations)

This paper contains 20 sections, 18 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Helly-type theorems and $d$-collapsibility
  3. Separated $d$-intervals
  4. Abstract convexity spaces and main results
  5. Helly related notions and main results
  6. Preliminaries
  7. Proofs of Theorem \ref{['theorem: colorful helly for k-intersecting']} and Theorem \ref{['theorem: fractional helly for k-intersecting']}
  8. Proof of Theorem \ref{['theorem: colorful helly for k-intersecting']}
  9. Proof of Theorem \ref{['theorem: fractional helly for k-intersecting']}
  10. Proof of Theorem \ref{['theorem: collapsibility']}
  11. Proof of Lemma \ref{['1cpq']}
  12. Proof of Lemma \ref{['2cpq']}
  13. Proof of Lemma \ref{['lemma: partial colorful Helly number']} and Lemma \ref{['lemma: colorful fractional Helly number']}
  14. Proofs of first three items of Theorem \ref{['theorem: Helly-type theorems']}
  15. Proof of \ref{['item1']} of Theorem \ref{['theorem: Helly-type theorems']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $K$ be a $d$-collapsible simplicial complex with the set of vertices $N=N_1 \sqcup \dots \sqcup N_{d+1}$ divided into $d+1$ disjoint subsets of sizes $|N_i|=n_i$, respectively. If $K$ contains at least $\alpha n_1 \dots n_{d+1}$ colorful $d$-faces, where $\alpha \in (0,1]$, that is, faces $\sigm

Theorems & Definitions (37)

  • Theorem 1.1: The optimal colorful fractional Helly theorem for $d$-collapsible complexes bulavka2021optimal
  • Definition 1.2
  • Theorem 1.3: Tardos tardos1995transversals, Kaiser kaiser1997transversals
  • Definition 1.4
  • Theorem 1.5
  • Definition 2.1: Radon partition and Radon number
  • Definition 2.2: Helly number
  • Definition 2.3: Colorful Helly number
  • Definition 2.4: Fractional Helly number
  • Definition 2.5: Colorful fractional Helly theorem
  • ...and 27 more