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Dimensional colorful Helly theorems and topological variants — arXiv:2609.36242

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.36242
License (as archived): CC0-1.0
Archived: 2026-10-07T18:33:31.237Z
SHA-256 of the archived text: 2e8f73ad6e6c974bf9e4e1270051c8c5ef424dfd3c7e6663f26dd6012bed45d9
Stable URL of this page: https://metraipolis.com/documento/dc_fc91dfdbe68f

How to cite

Dimensional colorful Helly theorems and topological variants — arXiv:2609.36242 — arXiv (en), https://arxiv.org/abs/2609.36242 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_fc91dfdbe68f (Metraipolis archive dc_fc91dfdbe68f, sha256 2e8f73ad6e6c974bf9e4e1270051c8c5ef424dfd3c7e6663f26dd6012bed45d9)

BibTeX

@misc{dcfc91dfdbe68f,
  title = {Dimensional colorful Helly theorems and topological variants — arXiv:2609.36242},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.36242},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_fc91dfdbe68f; sha256 2e8f73ad6e6c974bf9e4e1270051c8c5ef424dfd3c7e6663f26dd6012bed45d9},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.36242. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We prove a dimensional strengthening of the colorful Helly theorem. Let $\mathcal{C}_1,\dots,\mathcal{C}_{d+1}$ be finite nonempty families of convex sets in $\mathbb{R}^d$. If $C_1\cap\cdots\cap C_{d+1}\neq\varnothing$ for every choice of $C_i\in\mathcal{C}_i$, then $\sum_{i=1}^{d+1}\dim(\bigcap\mathcal{C}_i)\geq0$, where $\dim\varnothing=-1$. 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-30 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.36242 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T17:59:19.196Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-30?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).