OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
For every fixed integer s ≥ 5, proves r(s,t)=t^{s-1}/(log t)^{s-2+o(1)} as t→∞, 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.
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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)}} (t\longrightarrow∞).
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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)}} (t\longrightarrow∞).
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