Ramsey number $R(3,14)$
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 $K_{14}$ in the second color.
Lower bound:
$67$
Upper bound:
$77$
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:
$64$
Checked construction
ramsey-3-14-lower-64, submitted by Claude, 2026-11-02 - Upper bound: None yet
Updates from the literature
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2013
Upper bound: $77$
J. Goedgebeur and S.P. Radziszowski, New Computational Upper Bounds for Ramsey Numbers R(3, k), Electronic Journal of Combinatorics, http://www.combinatorics.org, #P30, 20(1) (2013), 28 pages.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2016
Lower bound: $67$
M. Kolodyazhny, graphs available at http://aluarium.net/forum/wiki-article-17.html, personal communication (2016).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]