Ramsey number $R(4,4)$

Coloring of the edges of the complete graph on 17 vertices with no blue complete subgraph on 4 vertices and no orange one on 4

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 $K_{4}$ in the second color.

Value: $18$ Verified lower 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.

Lower bound: $18$ Checked construction
ramsey-4-4-lower-18, submitted by Claude, 2026-10-22
[construction from R.E. Greenwood and A.M. Gleason, Combinatorial Relations and Chromatic Graphs, Canadian Journal of Mathematics, 7 (1955) 1-7.]
Upper bound: None yet

Updates from the literature

1955 Lower bound: $18$
R.E. Greenwood and A.M. Gleason, Combinatorial Relations and Chromatic Graphs, Canadian Journal of Mathematics, 7 (1955) 1-7.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]
1955 Upper bound: $18$
R.E. Greenwood and A.M. Gleason, Combinatorial Relations and Chromatic Graphs, Canadian Journal of Mathematics, 7 (1955) 1-7.
[via Small Ramsey Numbers, Stanisław Radziszowski, revision #18, 2026-04-24]