Sharp logarithmic exponents for off-diagonal Ramsey numbers

For every fixed integer s ≥ 5, proves r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)} as t\to\infty, determining the logarithmic exponent and matching the classical upper bound at that scale. Here r(s,t) is the least number of vertices forcing an s-clique or a t-vertex independent set. (Lean)

Manuscripts (2)

The sharp logarithmic exponent of r(5,t)

September 24, 2026 · source · cite as \cite{OAI:The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026}

We determine the sharp logarithmic exponent of the off-diagonal Ramsey number r(5,t):

\displaystyle r(5,t)=\frac{t^4}{(\log t)^{3+o(1)}} \qquad (t\longrightarrow\infty).

Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers

September 24, 2026 · source · cite as \cite{OAI:Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026}

For every fixed integer s ≥ 6, we determine the sharp logarithmic exponent of the off-diagonal Ramsey number:

\displaystyle r(s,t)=\frac{t^{s-1}}{(\log t)^{s-2+o(1)}} \qquad (t\longrightarrow\infty).

Result 170 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.