Dimensional colorful Helly theorems and topological variants

Abstract

We prove a dimensional strengthening of the colorful Helly theorem. Let 𝒞₁,…,𝒞_d+1 be finite nonempty families of convex sets in ℝ^d. If C₁∩⋯∩ C_d+1≠∅ for every choice of C_i∈𝒞_i, then ∑_i=1^d+1dim(⋂𝒞_i)≥0, where dim∅=−1. We also obtain a dimensional strengthening of a theorem of Kim and Lew, in which the intersections are taken over unions of color classes. For simplicial complexes, we introduce the link-Leray dimension, defined as the Leray number of the link for a face and as −1 for a nonface. Using this invariant, we prove a matroidal extension of the dimensional colorful Helly theorem for d-Leray complexes. We further establish a corresponding strengthening of the topological Kim--Lew theorem in two ranges of parameters. As a special case, we recover the topological colorful Helly theorem of Kalai and Meshulam. Finally, we construct counterexamples to the unrestricted topological extension, even under a stronger local condition.

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BibTeX RIS

BibTeX

@misc{rao2026dimensional,
  title = {{Dimensional colorful Helly theorems and topological variants}},
  author = {Wei Rao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36242},
  eprint = {2609.36242},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36242},
}

RIS

TY  - UNPB
TI  - Dimensional colorful Helly theorems and topological variants
AU  - Wei Rao
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36242
UR  - https://arxiv.org/abs/2609.36242
AB  - We prove a dimensional strengthening of the colorful Helly theorem. Let 𝒞_1,…,𝒞_d+1 be finite nonempty families of convex sets in ℝ^d. If C_1∩⋯∩ C_d+1≠∅ for every choice of C_i∈𝒞_i, then ∑_i=1^d+1dim(⋂𝒞_i)≥0, where dim∅=−1. We also obtain a dimensional strengthening of a theorem of Kim and Lew, in which the intersections are taken over unions of color classes. For simplicial complexes, we introduce the link-Leray dimension, defined as the Leray number of the link for a face and as −1 for a nonface. Using this invariant, we prove a matroidal extension of the dimensional colorful Helly theorem for d-Leray complexes. We further establish a corresponding strengthening of the topological Kim--Lew theorem in two ranges of parameters. As a special case, we recover the topological colorful Helly theorem of Kalai and Meshulam. Finally, we construct counterexamples to the unrestricted topological extension, even under a stronger local condition.
ER  -

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