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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)}

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.

Manuscripts in this family · 2

Sep 24, 2026·paper.pdf

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).

Full abstract

Sep 24, 2026·paper.pdf

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).

Full abstract