Branko Grunbaum in Many Dimensions — Question 1

Definitions and mathematical statement

The supplied record defines simplicity explicitly but leaves the meanings of a Venn diagram and symmetry implicit. The following formulation uses a planar interpretation with finite curve intersections and rotational symmetry; these conventions require confirmation against the source.

Let $n\geq 2$ be an integer. Consider distinct Jordan curves $C_1,\ldots,C_n\subset{\mathbb{R}}^2$, with each pair meeting in finitely many points. Write $D_i$ for the bounded component of ${\mathbb{R}}^2\setminus C_i$. For each subset $S\subseteq\{1,\ldots,n\}$, define \[ U_S=\left\{x\in{\mathbb{R}}^2\setminus\bigcup_{i=1}^n C_i:\ x\in D_i\ \Longleftrightarrow\ i\in S\text{ for every }i\right\}. \] The family is a Venn diagram if every $U_S$ is nonempty and connected. It is simple if \[ \#\{i:x\in C_i\}\leq 2\qquad\text{for every }x\in{\mathbb{R}}^2. \] Here symmetric means that, after relabeling, there is a point $o\in{\mathbb{R}}^2$ such that the rotation $\rho$ about $o$ through angle $2\pi/n$ satisfies \[ \rho(C_i)=C_{i+1}\quad(1\leq i<n),\qquad \rho(C_n)=C_1. \] Question (Kahle, 2019, Question 1). For every prime integer $n$, does there exist a simple, symmetric Venn diagram consisting of $n$ curves, under these conventions?

Short English statement

Does every prime number $n$ admit a planar Venn diagram with $n$ curves, all $2^n$ membership regions nonempty and connected, no point on more than two curves, and rotation through $2\pi/n$ cyclically permuting the curves? The rotational interpretation and curve conventions require source confirmation.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $800794$, exact source problem code catalogue entry; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry attributes the question to Kahle (2019) and labels it open; this label is supplied metadata, not an independently verified assessment.
• Kahle, Branko Grünbaum in Many Dimensions: Open Problems (2019), Question 1. Supplied source URL: https://arxiv.org/abs/1901.08622. The record reports access on 2026-10-16 and extraction by arxiv_latex; these provenance claims are supplied and have not been independently verified.