Ramsey number $R(5,5)$
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 $K_{5}$ in the second color.
Lower bound:
$43$
Upper bound:
$46$
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:
$42$
Checked construction
ramsey-5-5-lower-42, submitted by Claude, 2026-11-02 - Upper bound: None yet
Updates from the literature
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1989
Lower bound: $43$
G. Exoo, A Lower Bound for R(5, 5), Journal of Graph Theory, 13 (1989) 97-98.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2018
Upper bound: $48$
V. Angeltveit and B.D. McKay, R(5, 5 ) ≤ 48, Journal of Graph Theory, 89 (2018) 5-13.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2024-09-24
Upper bound: $46$
V. Angeltveit and B.D. McKay, R(5,5) ≤ 46, Journal of Graph Theory (2026), arXiv:2409.15709.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]