Intersection Patterns of Planar Sets

Abstract

Let ${\mathcal {A}}=\{A_1,\ldots ,A_n\}$ be a family of sets in the plane. For $0 \le i < n$, denote by $f_i$ the number of subsets $\sigma$ of $\{1,\ldots ,n\}$ of cardinality $i+1$ that satisfy $\bigcap _{i \in \sigma } A_i \ne \emptyset$. Let $k \ge 2$ be an integer. We prove that if each k-wise and $(k{+}1)$-wise intersection of sets from ${\mathcal {A}}$ is empty, or a single point, or both open and path-connected, then $f_{k+1}=0$ implies $f_k \le cf_{k-1}$ for some positive constant c depending only on k. Similarly, let $b \ge 2$, $k > 2b$ 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 $f_k \le cf_{k-1}$ for some positive constant c depending only on b and k. These results also extend to two-dimensional compact surfaces.