Nine Ramsey Number Lower-Bound Records
Improved lower bounds for nine classical Ramsey numbers, including , , and seven records up to , found by AlphaEvolve-discovered search algorithms.
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Ramsey theory
- Posed by
- —
- Year posed
- —
- Years open
- —
- Solved
- 2026-03-10
- Model
- AlphaEvolve
- Vendor
- Google DeepMind
- Collaborators
- Ansh Nagda, Prabhakar Raghavan, Abhradeep Thakurta
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Records, not resolutions: the exact values of these Ramsey numbers remain unknown.
What the AI did
AlphaEvolve, an LLM-based code-mutation system, ran a meta-search that discovered the search algorithms which produced the record constructions; Gemini 3 summarized the generated code for the paper.
Verification
Lower-bound constructions are finite certificates checkable by machine; no independent verification has been published yet.
Source
- PaperarXiv
What was checked: Nine lower-bound witnesses claimed in Nagda, Raghavan & Thakurta, "Reinforced Generation of Combinatorial Structures: Ramsey Numbers" (arXiv:2603.09172): R(3,13)≥61, R(3,18)≥100, and seven R(4,k) records — R(4,13)≥139, R(4,14)≥148, R(4,15)≥159, R(4,16)≥174, R(4,18)≥209, R(4,19)≥219, R(4,20)≥237.
Methodology: The nine witness graphs were fetched directly from the authors' public GitHub repository, converted to a standard edge-list format, and checked by two independently implemented programs that each confirm, from the raw adjacency data alone, that no monochromatic clique of the claimed size exists in either color. One checker was written from the mathematical definition of a Ramsey lower-bound witness only, without sight of the other checker's code.
Results: Both independently written checkers confirm all nine witnesses: 9/9 PASS on each, zero failures, zero mismatches, zero disagreements between the two checkers on any of the nine cases.
Runtime: one command re-verifies all nine witnesses in about four minutes — Python 3 stdlib only, no solver dependencies; the evidence repo's README opens with the exact quickstart. (That is the definition-blind checker. The other checker is the slow corroborating leg — up to 42 hours on the largest witness — and its complete logs ship alongside; don't use it for a spot-check.)
Literature status: Five of the nine records — R(3,13), R(3,18), R(4,13), R(4,14), R(4,15) — already appear in the Electronic Journal of Combinatorics' Dynamic Survey DS1 (revision 18) as the current accepted lower bounds. The remaining four — R(4,16), R(4,18), R(4,19), R(4,20) — are not yet reflected in either of DS1's relevant tables as of this check.
Prior verification: An arXiv search and a general web search found no other independently published verification of these nine values, and no superseding bound, as of 2026-08-13 (the staging sweep of 2026-08-11, re-run in full on posting day: same result).
Evidence: https://github.com/ZestyWombat854/ramsey-lower-bound-records-alphaevolve-verification