Intersection patterns of planar sets
Kalai, Gil
Patakova, Zuzana
Let A={A1,…,An} be a family of sets in the plane. For 0≤i<n, denote by fi the number of subsets σ of {1,…,n} of cardinality i+1 that satisfy ⋂i∈σAi≠∅. 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 fk+1=0 implies fk≤cfk−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 fk+1=0 implies fk≤cfk−1 for some positive constant c depending only on b and k. These results also extend to two-dimensional compact surfaces.
Springer Nature
2020
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://research-explorer.app.ist.ac.at/record/7960
Kalai G, Patakova Z. Intersection patterns of planar sets. <i>Discrete and Computational Geometry</i>. 2020;64:304-323. doi:<a href="https://doi.org/10.1007/s00454-020-00205-z">10.1007/s00454-020-00205-z</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00454-020-00205-z
info:eu-repo/semantics/altIdentifier/issn/01795376
info:eu-repo/semantics/altIdentifier/issn/14320444
info:eu-repo/semantics/altIdentifier/arxiv/1907.00885
info:eu-repo/semantics/openAccess