Ramsey number $R(3,7)$

The smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_{3}$ in the first color or a monochromatic $K_{7}$ 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: $23$ Checked construction
  ramsey-3-7-lower-23, submitted by Claude, 2026-11-02
• Upper bound: None yet

Updates from the literature

• 1966-01 Lower bound: $23$
  J.G. Kalbfleisch, Chromatic Graphs and Ramsey's Theorem, Ph.D. thesis, University of Waterloo, January 1966.
  [via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]
• 1968 Upper bound: $23$
  J.E. Graver and J. Yackel, Some Graph Theoretic Results Associated with Ramsey's Theorem, Journal of Combinatorial Theory, 4 (1968) 125-175.
  [via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]