Ramsey number $R(5,10)$
The smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_{5}$ in the first color or a monochromatic $K_{10}$ in the second color.
Lower bound:
$149$
Upper bound:
$381$
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
-
1994
Upper bound: $442$
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] -
2015
Lower bound: $149$
G. Exoo and M. Tatarevic, New Lower Bounds for 28 Classical Ramsey Numbers, Electronic Journal of Combinatorics, http:// www.combinatorics.org, #P3.11, 22(3) (2015), 12 pages. Graphs available at the journal site and at http://cs.indstate.edu/ge/RAMSEY/ExTa.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2020
Upper bound: $381$
V. Angeltveit and B.D. McKay, personal communication (2019-2024).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]