Ramsey number $R(3,10)$
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 $K_{10}$ in the second color.
Lower bound:
$40$
Upper bound:
$41$
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:
$39$
Checked construction
ramsey-3-10-lower-39, submitted by Claude, 2026-11-02 - Upper bound: None yet
Updates from the literature
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1989
Lower bound: $40$
G. Exoo, On Two Classical Ramsey Numbers of the Form R(3, n), SIAM Journal on Discrete Mathematics, 2 (1989) 488-490.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2024
Upper bound: $41$
V. Angeltveit, R(3, 10) ≤ 41, Electronic Journal of Combinatorics 32(4) (2025) #P4.30.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]