Ramsey number $R(5,7)$
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 $K_{7}$ in the second color.
Lower bound:
$80$
Upper bound:
$133$
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.
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Lower bound:
$71$
Checked construction
ramsey-5-7-lower-71, submitted by Claude, 2026-11-02 - Upper bound: None yet
Updates from the literature
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1993
Upper bound: $143$
Huang Yi Ru and Zhang Ke Min, A New Upper Bound Formula on Ramsey Numbers, Journal of Shanghai University, Natural Science, 7 (1993) 1-3.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
1997
Lower bound: $80$
N.J. Calkin, P. Erdős and C.A. Tovey, New Ramsey Bounds from Cyclic Graphs of Prime Order, SIAM Journal on Discrete Mathematics, 10 (1997) 381-387.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2020
Upper bound: $133$
V. Angeltveit and B.D. McKay, personal communication (2019-2024).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]