Ramsey number $R(7,13)$
The smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_{7}$ in the first color or a monochromatic $K_{13}$ in the second color.
Lower bound:
$511$
Upper bound:
$6653$
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
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2004
Lower bound: $511$
Xu Xiaodong, Xie Zheng and S.P. Radziszowski, A Constructive Approach for the Lower Bounds on the Ramsey Numbers R(s, t), Journal of Graph Theory, 47 (2004) 231-239.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2007
Upper bound: $10578$
(10,578)
Huang Yi Ru, Wang Yuandi, Sheng Wancheng, Yang Jiansheng, Zhang Ke Min and Huang Jian, New Upper Bound Formulas with Parameters for Ramsey Numbers, Discrete Mathematics, 307 (2007) 760-763.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2020
Upper bound: $6653$
V. Angeltveit and B.D. McKay, personal communication (2019-2024).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]