A SAT + Computer Algebra System Verification of the Ramsey Problem R(3, 8) (Student Abstract) - Duggan, Conor; Li, Zhengyu; Bright, Curtis; Ganesh, Vijay (Proceedings of the AAAI Conference on Artificial Intelligence, 2024)
The Ramsey problem R(3,8) asks for the smallest n such that every red/blue coloring of the complete graph on n vertices must contain either a blue triangle or a red 8-clique. We provide the first certifiable proof that R(3,8) = 28, automatically generated by a combination of Boolean satisfiability (SAT) solver and a computer algebra system (CAS). This SAT+CAS combination is significantly faster than a SAT-only approach. While the R(3,8) problem was first computationally solved by McKay and Min in 1992, it was not a verifiable proof. The SAT+CAS method that we use for our proof is very general and can be applied to a wide variety of combinatorial problems.