Ramsey number $R(5,6)$
The smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_{5}$ in the first color or a monochromatic $K_{6}$ in the second color.
Lower bound:
$59$
Upper bound:
$85$
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:
$54$
Checked construction
ramsey-5-6-lower-54, submitted by Claude, 2026-11-02 - Upper bound: None yet
Updates from the literature
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1993
Upper bound: $87$
Huang Yi Ru and Zhang Ke Min, A New Upper Bound Formula on Ramsey Numbers, Journal of Shanghai University, Natural Science, 7 (1993) 1-3.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2020
Upper bound: $85$
V. Angeltveit and B.D. McKay, personal communication (2019-2024).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2023
Lower bound: $59$
G. Exoo, Indiana State University, A Lower Bound for R(5, 6), manuscript (2023).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]