Math Conjectures

Multicolor Triangle Ramsey Growth

OPENLandmarkOpen problemProposed c. 1970 · Standard version

Canonical statement

Let Rk(3)R_k(3) be the least NN such that every coloring of the edges of KNK_N with kk colors contains a monochromatic triangle. Determine its exponential order, in particular whether
Rk(3)=kΘ(k)(k→∞). R_k(3)=k^{\Theta(k)}\qquad(k\to\infty).
View source LaTeX
Let \(R_k(3)\) be the least \(N\) such that every coloring of the edges of \(K_N\) with \(k\) colors contains a monochromatic triangle. Determine its exponential order, in particular whether
\[
  R_k(3)=k^{\Theta(k)}\qquad(k\to\infty).
\]
GRAPH-027:[Radziszowski2026Ramsey][OpenAI2026TenAdvances]
Graph theoryCombinatorics

Notes

Here Rk(3)R_k(3) is the diagonal kk-color Ramsey number for triangles, not the two-color finite number R(5,5)R(5,5) in GRAPH−019GRAPH-019. The longstanding asymptotic question asks whether its growth is kΘ(k)k^{\Theta(k)}, as tracked in modern Ramsey surveys [Radziszowski2026Ramsey]. A same-day August 1 manuscript claims the missing matching lower scale [OpenAI2026TenAdvances]; it is recorded as Grade C until independently checked.

Proof-claim watch (1)

Multicolor triangle Ramsey growth

Grade C

Full asymptotic-order claim awaiting independent verification.

References (2)

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