Bounding Helly numbers via Betti numbers

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F17%3A10368790" target="_blank" >RIV/00216208:11320/17:10368790 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/978-3-319-44479-6" target="_blank" >http://dx.doi.org/10.1007/978-3-319-44479-6</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/978-3-319-44479-6" target="_blank" >10.1007/978-3-319-44479-6</a>

Result language
angličtina
Original language name
Bounding Helly numbers via Betti numbers
Original language description
We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b, d) such that the following holds. If F is a finite family of subsets of ℝd such that β i(intersection of G)<=b for any proper subset G of F and every 0 <= i <= ceil(d/2)- 1 then F has Helly number at most h(b, d). Here β i denotes the reduced ℤ2 -Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach .
Czech name
—
Czech description
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Project
<a href="/en/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Center of excellence - Institute for theoretical computer science (CE-ITI)</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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