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arXiv · 2501.03207

Helly-type theorems for separated $d$-intervals

Wei Rao
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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.

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Wei Rao. 2025-05-22. Helly-type theorems for separated $d$-intervals. https://arxiv.org/abs/2501.03207

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