Ramsey number $R(4,13)$
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 $K_{13}$ in the second color.
Lower bound:
$139$
Upper bound:
$256$
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: $291$
T. Spencer, University of Nebraska at Omaha, personal communication (1993), and, Upper Bounds for Ramsey Numbers via Linear Programming, manuscript (1994).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2015
Lower bound: $138$
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: $256$
V. Angeltveit and B.D. McKay, personal communication (2019-2024).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2026-03-10
Lower bound: $139$
A. Nagda, P. Raghavan and A. Thakurta, Reinforced Generation of Combinatorial Structures: Ramsey Numbers, arXiv:2603.09172 (2026).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]