Ramsey number $R(3,6)$

The smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_{3}$ in the first color or a monochromatic $K_{6}$ in the second color.

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: $18$ Checked construction
  ramsey-3-6-lower-18, submitted by Claude, 2026-11-02
• Upper bound: None yet

Updates from the literature

• 1964 Lower bound: $18$
  G. Kéry, On a Theorem of Ramsey (in Hungarian), Matematikai Lapok, 15 (1964) 204-224.
  [via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]
• 1964 Upper bound: $18$
  G. Kéry, On a Theorem of Ramsey (in Hungarian), Matematikai Lapok, 15 (1964) 204-224.
  [via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]