Let \(R_k\)(3) be the least n such that any k-colouring of the edges of \(K_n\) yields a monochromatic triangle. Erdős asked how fast \(R_k\)(3) grows; the conjectured lower-bound mechanism (probabilistic) gave roughly c·k! growth. Erdős problem #183 asks for a superexponential construction.
Ramsey numbers are notoriously resistant: even R(5,5) is unknown between 43 and 48. The multicolour triangle function was pinned between exponential and factorial bounds for decades, with improvements measured in constant factors.
Resolved by OpenAI's Astra model, 1 August 2026: a superexponential lower bound for multicolour triangle Ramsey numbers, settling Erdős problem #183. Listed among the ten advances alongside resolutions of Erdős problems #146 and #180 (compactness/degeneracy in extremal graph theory) — three catalogue entries of Bloom's database retired in one batch.
Context: the two-colour case is the textbook \(R(3,3) = 6\); for fixed \(k\) the probabilistic method gives lower bounds on roughly the \(k!\) scale, and the question was whether explicit or semi-explicit colourings can beat every exponential. A genuine superexponential construction must exhibit, for each \(k\), a \(k\)-colouring of \(K_N\) with \(N\) growing faster than any exponential in \(k\) and no monochromatic triangle in any colour. Both halves are finite combinatorial checks — vertex count asymptotics plus triangle-freeness per colour class — which is the part of such claims most amenable to mechanised certificates of the kind accompanying this batch.
Verified by curator — see the API for full claim provenance.
