Ramsey number $R(10,10)$
The smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_{10}$ in the first color or a monochromatic $K_{10}$ in the second color.
Lower bound:
$798$
Upper bound:
$16064$
(16,064)
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: None yet
- Upper bound: None yet
Updates from the literature
-
1987
Lower bound: $798$
R. Mathon, Lower Bounds for Ramsey Numbers and Association Schemes, Journal of Combinatorial Theory, Series B, 42 (1987) 122-127.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2003
Upper bound: $23556$
(23,556)
Lingsheng Shi, Upper Bounds for Ramsey Numbers, Discrete Mathematics, 270 (2003) 251-265.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2020
Upper bound: $16064$
(16,064)
V. Angeltveit and B.D. McKay, personal communication (2019-2024).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]