Ramsey number $R(4,12)$
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 $K_{12}$ in the second color.
Lower bound:
$128$
Upper bound:
$210$
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: $238$
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] -
1998
Lower bound: $128$
Su Wenlong, Luo Haipeng and Li Qiao, New Lower Bounds of Classical Ramsey Numbers R(4, 12), R(5, 11) and R(5, 12), Chinese Science Bulletin, 43, 6 (1998) 528.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2020
Upper bound: $210$
V. Angeltveit and B.D. McKay, personal communication (2019-2024).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]