Can the five Ramsey numbers still listed as open in the field's canonical survey be determined exactly from results already published?
Yes. The April 24, 2026 revision of Small Ramsey Numbers lists five values in one well-studied family as unresolved. All five have exact answers.
Theorems from 1991 and 2002 reduce the questions to specific combinatorial objects. Papers from 2015–2021 constructed those objects without stating the Ramsey consequences. Connecting the two bodies of work gives:
Together with what the survey already records, these close the family's entire stated range:
What each number asks
Colour every edge of a complete graph on N vertices red or blue. The Ramsey number R(G, H) is the smallest N at which a red copy of G or a blue copy of H becomes unavoidable; no colouring escapes both. K2,n is the complete bipartite graph joining two vertices to n others, and it has a concrete meaning here: a red K2,17 exists exactly when two vertices share at least 17 common red neighbours. So R(K2,17, K2,17) = 66 says that among any 66 people, some pair has 17 common friends or 17 people who are strangers to both. Among 65, friendship can be arranged so that no pair has either.
Why the five gaps survived
In 1991, Exoo, Harborth, and Mengersen proved that R(K2,n, K2,n) = 4n − 2 precisely when a strongly regular graph with a specific parameter set exists. In 2002, Lortz and Mengersen proved R(K2,n−1, K2,n) ≤ 4n − 4, with equality whenever a symmetric Hadamard matrix of order 4n − 4 exists. Both theorems convert a Ramsey question into an existence question about a single combinatorial object.
Then the objects arrived. Symmetric Hadamard matrices of orders 116 and 172 were constructed in 2015, order 156 in 2017, and order 188 in 2018. In 2021, Gritsenko constructed a strongly regular graph with parameters (65, 32, 15, 16), exactly the set the 1991 characterization demands at n = 17. As far as our searches found, none of the construction papers draw the Ramsey consequence, and the survey's April 2026 revision still lists all five values as open.
Five exact answers
Each equality is the two halves meeting. Gritsenko's graph, coloured red with its non-edges blue, gives every pair of its 65 vertices exactly 15 or 16 common neighbours in each colour, never the 17 a monochromatic K2,17 needs. That matches the known upper bound of 66 and settles the survey's first open diagonal case. For the asymmetric values, deleting the first row and column of each normalized Hadamard matrix yields a colouring in which one colour never reaches n − 1 shared neighbours and the other never reaches n, meeting the Lortz–Mengersen bound from below.
The survey names n = 30, 40, 44, and 48 as the only unresolved cases below 59; these are exactly the four now closed, which is what completes the 4n − 4 classification for 3 ≤ n ≤ 58.
Why the certificates matter
A claim that lives across five papers deserves more than citations. The published record includes finite certificates for all five values and a deterministic verifier that uses only the Python standard library. It checks the full 65-vertex adjacency matrix (symmetry, degrees, and every pair's codegree in both colours), then rebuilds all four Hadamard matrices from the published difference families and checks every inner product and codegree condition. No step involves search or randomness. Anyone can rerun it and compare digests.
Concorde, an independent research system, surfaced the diagonal consequence first. Beyond, our research system, then expanded the reconciliation to the four remaining values and helped prepare the verifier, the prior-art audit, and the manuscript. Research systems are not authors; the authors made the final scientific judgments and accept full responsibility.
What this does and doesn't establish
The claim is deliberately narrow. We constructed no new strongly regular graph, no new Hadamard matrix, and no new Ramsey method; every proof is a short, careful substitution of a published object into a published theorem. The contribution is the previously unrecorded reconciliation, the exact classification it completes, and the reproducible verification. No earlier public statement of any of the five equalities was found as of August 10, 2026, after searches across notations, parameters, and citation indexes. That does not establish that nobody ever knew them privately.
Independent correctness and openness review passed. Two separate significance reviews rated the result below our strict standalone-publication bar because the derivations are direct substitutions. We published the exact-determination record with that disagreement preserved because five entries the field's live survey explicitly lists as open, including its first open diagonal case, are now exact.
Read it, run it
Archived record (manuscript and certificates): doi.org/10.5281/zenodo.21878986
Verifier and certificates: github.com/Nth-Research-Collective/research. Run python3 projects/ramsey-k2n/verification/verify.py from the repository root.
If you know of an earlier statement of any of these values, please send it to research@nthresearch.org.