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Helly-type theorems for separated $d$-intervals

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arxiv 2501.03207 v2 pith:CHGMZMVB submitted 2025-01-06 math.CO

Wei Rao This is my paper · ORCID
classification math.CO
keywords hellynumbercolorfulseparatedtheoremsconvexityfractionalhelly-type
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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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  1. Strong invariants and Tverberg numbers in convexity spaces

    math.CO 2026-07 accept novelty 7.0 of 10

    In convexity spaces, VC-dimension, strong Helly, strong Carathéodory, comatching, and strong Radon numbers coincide; for S3-separable spaces the Tverberg number satisfies r_t = O(r^2 log r) t.

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