Bounding Radon Numbers via Betti Numbers
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10488672" target="_blank" >RIV/00216208:11320/24:10488672 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=u2vtUItR3p" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=u2vtUItR3p</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1093/imrn/rnae056" target="_blank" >10.1093/imrn/rnae056</a>
Alternative languages
Result language
angličtina
Original language name
Bounding Radon Numbers via Betti Numbers
Original language description
We prove general topological Radon-type theorems for sets in $mathbb R<^>{d}$ or on a surface. Combined with a recent result of Holmsen and Lee, we also obtain fractional Helly theorem, and consequently the existence of weak $varepsilon $-nets as well as a $(p,q)$-theorem for those sets. More precisely, given a family ${mathcal{F}}$ of subsets of ${mathbb{R}}<^>{d}$, we will measure the homological complexity of ${mathcal{F}}$ by the supremum of the first $lceil d/2rceil $ reduced Betti numbers of $bigcap{mathcal{G}}$ over all nonempty ${mathcal{G}} subseteq{mathcal{F}}$. We show that if ${mathcal{F}}$ has homological complexity at most $b$, the Radon number of ${mathcal{F}}$ is bounded in terms of $b$ and $d$. In case that ${mathcal{F}}$ lives on a surface and the number of connected components of $bigcap mathcal G$ is at most $b$ for any $mathcal Gsubseteq mathcal F$, then the Radon number of ${mathcal{F}}$ is bounded by a function depending only on $b$ and the surface itself. For surfaces, if we moreover assume the sets in ${mathcal{F}}$ are open, we show that the fractional Helly number of $mathcal F$ is linear in $b$. The improvement is based on a recent result of the author and Kalai. Specifically, for $b=1$ we get that the fractional Helly number is at most three, which is optimal. This case further leads to solving a conjecture of Holmsen, Kim, and Lee about an existence of a $(p,q)$-theorem for open subsets of a surface.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GA22-19073S" target="_blank" >GA22-19073S: Combinatorial and computational complexity in topology and geometry</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
International Mathematics Research Notices
ISSN
1073-7928
e-ISSN
1687-0247
Volume of the periodical
2024
Issue of the periodical within the volume
11
Country of publishing house
GB - UNITED KINGDOM
Number of pages
19
Pages from-to
9482-9500
UT code for WoS article
001195357400001
EID of the result in the Scopus database
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