Erdős Problem #986 — For any fixed $k\geq 3$, $R(k,n) \gg \frac{n^{k-1}}{(\log n)^c}$ for some constant $c=c(k)>0$.

The conjecture asks whether, for every fixed $k\geq 3$, the off-diagonal Ramsey number satisfies the stated near-$n^{k-1}$ lower bound up to a logarithmic factor. It is attributed to Erdős, apparently from 1947.

Known results

• Spencer (1977) settled $k=3$.
• Mattheus and Verstraete (2023) proved $R(4,n)=n^{3+o(1)}$.
• Ajtai, Komlós, and Szemerédi proved the corresponding upper bound $R(k,n)\ll_k n^{k-1}/(\log n)^{k-2}$.
• Earlier general lower bounds were weaker for $k\geq 5$.

May 2026 claimed theorem

Bradač’s preprint claims that, for every fixed $k\geq 3$, $R(k,n)\gg n^{k-1}/(\log n)^{2k-4}$, which implies the conjecture. The preprint reports assistance from an unnamed internal OpenAI model; the mathematical claim remains unverified.

Current status (as of September 2026): Bradač’s preprint claims the bound for every fixed $k\geq 3$, but independent verification is still absent.