article published yes Gil Kalai author Zuzana Patakova author 48B57058-F248-11E8-B48F-1D18A9856A870000-0002-3975-1683 UlWa department 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 Nature2020 eng 01795376 14320444 1907.00885 00053732940000110.1007/s00454-020-00205-z 64304-323 G. Kalai and Z. Patakova, “Intersection patterns of planar sets,” <i>Discrete and Computational Geometry</i>, vol. 64. Springer Nature, pp. 304–323, 2020. G. Kalai, Z. Patakova, Discrete and Computational Geometry 64 (2020) 304–323. Kalai G, Patakova Z. 2020. Intersection patterns of planar sets. Discrete and Computational Geometry. 64, 304–323. 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> Kalai, G., &#38; Patakova, Z. (2020). Intersection patterns of planar sets. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href="https://doi.org/10.1007/s00454-020-00205-z">https://doi.org/10.1007/s00454-020-00205-z</a> Kalai, Gil, and Zuzana Patakova. “Intersection Patterns of Planar Sets.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2020. <a href="https://doi.org/10.1007/s00454-020-00205-z">https://doi.org/10.1007/s00454-020-00205-z</a>. Kalai, Gil, and Zuzana Patakova. “Intersection Patterns of Planar Sets.” <i>Discrete and Computational Geometry</i>, vol. 64, Springer Nature, 2020, pp. 304–23, doi:<a href="https://doi.org/10.1007/s00454-020-00205-z">10.1007/s00454-020-00205-z</a>. 79602020-06-14T22:00:50Z2023-08-21T08:26:34Z