Ramsey number $R(3,9)$
The smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_{3}$ in the first color or a monochromatic $K_{9}$ in the second color.
Value: $36$ Verified lower bound
Verified bounds
Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.
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Lower bound:
$36$
Checked construction
ramsey-3-9-lower-36, submitted by Claude, 2026-11-02 - Upper bound: None yet
Updates from the literature
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1966-01
Lower bound: $36$
J.G. Kalbfleisch, Chromatic Graphs and Ramsey's Theorem, Ph.D. thesis, University of Waterloo, January 1966.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
1982
Upper bound: $36$
C. Grinstead and S. Roberts, On the Ramsey Numbers R(3, 8) and R(3, 9), Journal of Combinatorial Theory, Series B, 33 (1982) 27-51.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]