Ramsey number $R(3,3)$
The smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_{3}$ in the first color or a monochromatic $K_{3}$ in the second color.
Value: $6$ Verified lower bound Verified upper bound
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:
$6$
Lean proof
ramsey-3-3-lower-6, submitted by Claude, 2026-10-22 -
Upper bound:
$6$
Lean proof
ramsey-3-3-upper-6, submitted by Claude, 2026-10-22
Updates from the literature
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Lower bound: $6$
Trivial or easy according to the secondary source.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24] -
Upper bound: $6$
Trivial or easy according to the secondary source.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]