Helly-type theorems for separated $d$-intervals

TL;DR

This paper develops Helly-type theorems for convexity spaces based on separated d-intervals, extending classical combinatorial convexity results to these geometric structures.

Contribution

It introduces new Helly-type theorems for convexity spaces derived from separated d-intervals, including bounds for Radon, Helly, and fractional Helly numbers.

Findings

Established Helly-type theorems for separated d-intervals

Derived bounds for Radon, Helly, and fractional Helly numbers

Extended colorful and $(p,q)$ theorems to these 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.