Verified Certificates via SAT and Computer Algebra Systems for the Ramsey R(3, 8) and R(3, 9) Problems

2025-02-09

AI Summary

This work addresses the long-standing lack of formal verification for the Ramsey numbers $R(3,8)$ and $R(3,9)$—i.e., the smallest $n$ such that every red/blue edge-coloring of $K_n$ contains either a blue triangle or a red $K_8$ (or $K_9$). We introduce the first verifiable parallel cube-and-conquer framework integrating SAT solvers with computer algebra systems (CAS), implemented via the MATHCHECK toolchain to ensure both correctness and completeness of exhaustive search. Our approach yields the first independent, machine-checkable formal certificates for $R(3,8)$, $R(3,9)$, and the symmetric case $R(8,3)$. Notably, $R(8,3)$ is solved in just 18.5 hours—dramatically outperforming conventional SAT-based methods, which time out after seven days. This work overcomes a critical bottleneck in the verifiability of combinatorial proofs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The Ramsey problem R(3, k) seeks to determine the smallest value of n such that any red/blue edge coloring of the complete graph on n vertices must either contain a blue triangle (3-clique) or a red clique of size k. Despite its significance, many previous computational results for the Ramsey R(3, k) problem such as R(3, 8) and R(3, 9) lack formal verification. To address this issue, we use the software MATHCHECK to generate certificates for Ramsey problems R(3, 8) and R(3, 9) (and symmetrically R(8, 3) and R(9, 3)) by integrating a Boolean satisfiability (SAT) solver with a computer algebra system (CAS). Our SAT+CAS approach significantly outperforms traditional SAT-only methods, demonstrating a significant improvement in runtime. For instance, our SAT+CAS approach solves R(8, 3) sequentially in 18.5 hours, while a SAT-only approach using the state-of-the-art CaDiCaL solver times out after 7 days. Additionally, in order to be able to scale to harder Ramsey problem like R(9, 3) we further optimized our SAT+CAS tool using a parallelized cube-and-conquer approach. Our results provide the first independently verifiable certificates for these Ramsey numbers, ensuring both correctness and completeness of the exhaustive search process of our SAT+CAS tool.

Problem

• Verifies Ramsey R(3,8) and R(3,9) solutions
• Integrates SAT solver with CAS for verification
• Provides first verifiable certificates for Ramsey numbers

Innovation

• Combines SAT solver with CAS
• Uses parallelized cube-and-conquer method
• Generates verifiable Ramsey problem certificates