Determine the open Ramsey numbers R(3,10), R(4,6), and R(5,5)

Determine the exact values of the Ramsey numbers R(3,10), R(4,6), and R(5,5), which remain unresolved.

Background

The paper develops a SAT+CAS methodology for producing independently checkable certificates for exhaustive computational searches in Ramsey theory and applies it to verify R(3,8)=28 and R(3,9)=36. The conclusion identifies determining R(3,10), R(4,6), and R(5,5) as future applications of SAT+CAS because these Ramsey numbers remain open. This is a concrete unresolved mathematical objective rather than a general methodological aspiration.

References

Future applications of SAT+CAS to Ramsey problems include the verification of all known Ramsey numbers and determining the values of $R(3,10)$, $R(4,6)$, or $R(5,5)$, which remain open problems.

— Verified Certificates via SAT and Computer Algebra Systems for the Ramsey R(3, 8) and R(3, 9) Problems  (2502.06055 - Li et al., 9 Feb 2025) in Section Conclusion