Verified Certificates via SAT and Computer Algebra Systems for the Ramsey R(3,8) and R(3,9) Problems
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
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 an improvement of several orders of magnitude in runtime. For instance, our SAT+CAS approach solves R(3,8) (resp., R(8,3)) sequentially in 59 hours (resp., in 11 hours), while a SAT-only approach using state-of-the-art CaDiCaL solver times out after 7 days. Additionally, in order to be able to scale to harder Ramsey problems R(3,9) and 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.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it