Intersection patterns of planar sets
Gil Kalai, Zuzana Patáková · arXiv (Cornell University) · 2019
Let $\mathcal A=\{A_1,\ldots,A_n\}$ be a family of sets in the plane. For $0 \leq i 2b$ be integers. We prove that if each $k$-wise or $(k+1)$-wise intersection of sets from $\mathcal A$ has at most $b$ path-connected components, which all are open, then $f_{k+1}=0$ implies $f_k \leq cf_{k-1}$ for some positive constant $c$ depending only on $b$ and $k$. These results also extend to two-dimensional compact surfaces.