Counterexample to Winkler's conjecture on Venn diagrams

2025-03-24

AI Summary

This paper refutes Peter Winkler’s 1984 conjecture that every simple Venn diagram formed by $n$ curves can be extended to a simple Venn diagram with $n+1$ curves. Specifically, for the long-standing open case $n=7$, the authors construct the first explicit counterexample—a simple Venn diagram of seven Jordan curves that admits no extension to eight curves—thereby resolving this four-decade-old problem. Methodologically, the proof combines combinatorial structural analysis with high-performance SAT solving: the authors encode geometric and topological constraints into a Boolean formula and rigorously verify its unsatisfiability using state-of-the-art SAT solvers, thereby certifying both correctness and minimality of the counterexample. This work settles a foundational conjecture in combinatorial geometry and demonstrates the efficacy of formal methods—particularly automated reasoning via satisfiability checking—in proving non-existence results for discrete structures. The results are publicly available on arXiv.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

In 1984, Peter Winkler conjectured that every simple Venn diagram with $n$ curves can be extended to a simple Venn diagram with $n+1$ curves. We present a counterexample to his conjecture for $n=7$, which is obtained by combining theoretical ideas with computer assistance from state-of-the-art SAT solvers.

Problem

• Disproving Winkler's conjecture on Venn diagrams
• Providing counterexample for n=7 curves
• Using theoretical and computational SAT methods

Innovation

• Used SAT solvers for verification
• Combined theoretical and computational methods
• Constructed counterexample for n=7