Intersection patterns of planar sets

TL;DR

This paper establishes bounds on the number of intersections of certain sizes in planar set families under specific connectivity conditions, extending results to surfaces.

Contribution

It proves new inequalities relating intersection counts in planar sets with topological constraints, generalizing previous intersection theory results.

Findings

If higher-order intersections are empty or simple, then lower-order intersection counts are bounded.

Conditions on intersection connectivity lead to inequalities between intersection counts.

Results extend to compact surfaces beyond the plane.

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.