Intersection patterns of planar sets

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10422276" target="_blank" >RIV/00216208:11320/20:10422276 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=vQyQAB5B4u" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=vQyQAB5B4u</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00454-020-00205-z" target="_blank" >10.1007/s00454-020-00205-z</a>

Result language
angličtina
Original language name
Intersection patterns of planar sets
Original language description
Let A = {A(1) , ... , A(n)} be a family of sets in the plane. For 0 <= i < n, denote by f(i) the number of subsets sigma of {1, ... , n} of cardinality i + 1 that satisfy boolean AND(i is an element of sigma) A(i) not equal theta. Let k >= 2 be an integer. We prove that if each k-wise and (k+1)-wise intersection of sets from A is empty, or a single point, or both open and path-connected, then f(k+1) = 0 implies f(k) <= cf(k-1) for some positive constant c depending only on k. Similarly, let b >= 2, k > 2b be integers. We prove that if each k-wise or (k+1)-wise intersection of sets from A has at most b path-connected components, which all are open, then f(k+1) = 0 implies f(k) <= cf(k-1) for some positive constant c depending only on b and k. These results also extend to two-dimensional compact surfaces.
Czech name
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Czech description
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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

Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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