Growth of multicolour triangle Ramsey numbers
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Let R_k(3) be the multicolour Ramsey number forcing a monochromatic triangle in a k-edge-coloured complete graph. The growth target is log R_k(3)=Θ(k log k), supported on the lower side by (c k^(1/3)/log k)^k.
Why it matters and what counts as progress
Why it matters: It distinguishes superexponential triangle-Ramsey growth from the previously possible exponential scale.
What counts as progress: Improve multicolour triangle-Ramsey constructions or upper bounds toward the k log k logarithmic scale, keeping the number of colours variable.
How the problem developed
- 1961 · Open problem [1]
Paul Erdős Posed the problem of determining the limit of the kth root of the multicolour triangle Ramsey number.
Romain Ageron, Paul Casteras, Thibaut Pellerin, Yann Portella, Arpad Rimmel, and Joanna Tomasik Improved exponential lower bounds through Schur-number constructions, giving R(3;k)≥380^(k/5)−O(1).
OpenAI internal model Announced a superexponential lower bound of the form R(3;k)≥(c k^(1/3)/log k)^k, which implies that the kth roots diverge to infinity.
AI contribution
Ten Advances in Mathematics and Theoretical Computer Science
A recursive k-edge-coloring of a complete graph without a monochromatic triangle gives Rₖ(3)≥(c k^(1/3)/log k)^k for all sufficiently large k. Thus log Rₖ(3)=Θ(k log k).
Sources
- Erdős Problems, history of problem 183
- Ageron et al., New lower bounds for Schur and weak Schur numbers
- OpenAI, Ten Advances in Mathematics release and attribution
- OpenAI, Lean formalization for the multicolour triangle Ramsey result
- OpenAI, Ten Advances in Mathematics, Chapter 9
- OpenAI, Ramsey formalization manifest
- Sienicki and coauthors, Chapter 9 human review
- Erdős Problem 183
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