Ramsey number $R(4,9)$
The smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_{4}$ in the first color or a monochromatic $K_{9}$ in the second color.
Lower bound:
$73$
Upper bound:
$105$
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.
-
Lower bound:
$69$
Checked construction
ramsey-4-9-lower-69, submitted by Claude, 2026-11-02 - Upper bound: None yet
Updates from the literature
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1994
Upper bound: $115$
J. Mackey, Combinatorial Remedies, Ph.D. thesis, Department of Mathematics, University of Hawaii, 1994.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2005
Lower bound: $73$
G. Exoo, Indiana State University, personal communication (2005-2006). Constructions available at http://cs.indstate.edu/ge/RAMSEY.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2020
Upper bound: $105$
V. Angeltveit and B.D. McKay, personal communication (2019-2024).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]