Ramsey number $R(4,5)$

The smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_{4}$ in the first color or a monochromatic $K_{5}$ 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: $25$ Checked construction
  ramsey-4-5-lower-25, submitted by Claude, 2026-11-02
• Upper bound: None yet

Updates from the literature

• 1965 Lower bound: $25$
  J.G. Kalbfleisch, Construction of Special Edge-Chromatic Graphs, Canadian Mathematical Bulletin, 8 (1965) 575-584.
  [via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]
• 1995 Upper bound: $25$
  B.D. McKay and S.P. Radziszowski, R(4, 5) = 25, Journal of Graph Theory, 19 (1995) 309-322.
  [via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]