Ramsey number $R(4,6)$
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 $K_{6}$ in the second color.
Lower bound:
$36$
Upper bound:
$40$
Verified bounds
Verified bounds come with a certificate whose evidence has been checked here: a formal proof in Lean, or an explicit construction checked by a program. An Attested bound comes with evidence that someone else checked with a published checker, but that hasn't been checked here yet. Submit a certificate.
-
Lower bound:
$34$
Checked construction
ramsey-4-6-lower-34, submitted by daniel-munro, 2026-11-02 -
Upper bound:
$50$
Lean proof
ramsey-4-6-upper-50, submitted by daniel-munro, 2026-11-11
Updates from the literature
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1997
Upper bound: $41$
B.D. McKay and S.P. Radziszowski, Subgraph Counting Identities and Ramsey Numbers, Journal of Combinatorial Theory, Series B, 69 (1997) 193-209.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2012
Lower bound: $36$
G. Exoo, On the Ramsey Number R(4, 6), Electronic Journal of Combinatorics, http://www.combinatorics.org, #P66, 19(1) (2012), 5 pages.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
2020
Upper bound: $40$
V. Angeltveit and B.D. McKay, personal communication (2019-2024).
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]