Erdős Problem #986

Erdős apparently posed the conjecture in 1947: for each fixed $sgeq 3$, the off-diagonal Ramsey number should be at least $k^{s-1}$ up to a polylogarithmic factor. Bradač’s 2026 work establishes this for all fixed $s$.

Known results

• Spencer (1977) settled the case $s=3$.
• Mattheus and Verstraete (2023) settled the case $s=4$.
• Ajtai, Komlós, and Szemerédi proved the upper bound $R(s,k)\ll_s k^{s-1}/(\log k)^{s-2}$.
• Earlier lower bounds were weaker; Bradač’s result improves the best bounds for $s\geq 5$.

May 2026 lower-bound theorem

Bradač proved that, for every fixed $s\geq 3$,

$$R(s,k)\gg \frac{k^{s-1}}{(\log k)^{2s-4}}.$$

This directly implies the requested bound. The preprint says the final improvement was made with assistance from an unnamed internal OpenAI model; no specific model is identified.

Current status (as of May 2026): The conjectured lower bound is proved for every fixed $s\geq 3$ by Bradač’s preprint, and the problem is resolved.