Determine exactly which prime numbers p admit a simple rotationally symmetric Venn diagram consisting of p Jordan curves on the sphere, invariant under a rotation through 2 pi/p that carries the curves to one another, such that all 2^p regions are present and connected and every crossing involves exactly two curves.

Existence of simple symmetric Venn diagrams

The problem asks which prime numbers of curves admit simple rotationally symmetric Venn diagrams. General rotationally symmetric diagrams exist for every prime number, but simplicity is known only for selected sizes.

Known results

Henderson, 1963: rotational symmetry requires the number of curves to be prime.
Griggs, Killian, and Savage, 2004: rotationally symmetric diagrams exist for every prime number of curves.
Mamakani and Ruskey, 2012: simple symmetric examples were found for $11$ curves; the known list later included $1,3,5,7,11,13$ .
A 2025 survey states that general simple symmetric existence remains open.

September 2026 construction

Chris Dzoba reports simple symmetric Venn diagrams with $17$ and $19$ curves, together with Lean $4$ formal certificates. This extends the known range beyond $13$ , but the report supplies examples at only two sizes and does not establish existence for every larger admissible size; the claim is unverified.

Current status (as of September 2026): examples are reported at $17$ and $19$ curves with Lean $4$ certificates, while existence of simple symmetric diagrams for all larger prime numbers remains open and the new claim is unverified.

Existence of simple symmetric Venn diagrams

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