Intersection patterns of planar sets

Abstract

Let $\mathcal{A}=\{A_1,\dots ,A_n\}$ be a family of sets in the plane. For $0\le i<n$, denote by $f_i$ the number of subsets $\sigma$ of $\{1,\dots ,n\}$ of cardinality $i+1$ that satisfy ${\bigcap }_{i\in \sigma }{A}_i\ne \varnothing$. 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 $f_k\le c{f}_{k-1}$ 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 $f_k\le c{f}_{k-1}$ for some positive constant $c$ depending only on $b$ and $k$. These results also extend to two-dimensional compact surfaces.