Bounding Radon Numbers via Betti Numbers

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Bounding Radon Numbers via Betti Numbers

Popis výsledku v původním jazyce

We prove general topological Radon-type theorems for sets in $mathbb R&lt;^&gt;{d}$ 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 ${mathbb{R}}&lt;^&gt;{d}$, 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.

Název v anglickém jazyce

Bounding Radon Numbers via Betti Numbers

Popis výsledku anglicky

We prove general topological Radon-type theorems for sets in $mathbb R&lt;^&gt;{d}$ 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 ${mathbb{R}}&lt;^&gt;{d}$, 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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA22-19073S" target="_blank" >GA22-19073S: Kombinatorická a výpočetní složitost v topologii a geometrii</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

International Mathematics Research Notices

ISSN

1073-7928

e-ISSN

1687-0247

Svazek periodika

2024

Číslo periodika v rámci svazku

11

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

19

Strana od-do

9482-9500

Kód UT WoS článku

001195357400001

EID výsledku v databázi Scopus

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