Intersection patterns of planar sets

Abstract: Let $\mathcal A={A_1,\ldots,A_n}$ be a family of sets in the plane. For $0 \leq i < n$, denote by $f_i$ the number of subsets $\sigma$ of ${1,\ldots,n}$ of cardinality $i+1$ that satisfy $\bigcap_{i \in \sigma} A_i \neq \emptyset$. Let $k \geq 2$ 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 $f_k \leq cf_{k-1}$ for some positive constant $c$ depending only on $k$. Similarly, let $b \geq 2, k > 2b$ 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 $f_k \leq cf_{k-1}$ for some positive constant $c$ depending only on $b$ and $k$. These results also extend to two-dimensional compact surfaces.