Sharp logarithmic exponents for off-diagonal Ramsey numbers at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Sharp logarithmic exponents for off-diagonal Ramsey numbers

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Sharp logarithmic exponents for off-diagonal Ramsey numbers. For every fixed integer s ≥ 5, proves $r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)}$ 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.

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LEVEL 1 The sharp logarithmic exponent of r(5,t)

released 2026-09-24 | 4 theorems · 18 lemmas · 21 proofs · 18,758 words | PLAY LEVEL 1 » (pdf)

We determine the sharp logarithmic exponent of the off-diagonal Ramsey number $r(5,t)$: $\displaystyle r(5,t)=\frac{t^4}{(\log t)^{3+o(1)}} \qquad (t\longrightarrow\infty).$

LEVEL 2 Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers

released 2026-09-24 | 2 theorems · 30 lemmas · 37 proofs · 24,298 words | PLAY LEVEL 2 » (pdf)

For every fixed integer s ≥ 6, we determine the sharp logarithmic exponent of the off-diagonal Ramsey number: $\displaystyle r(s,t)=\frac{t^{s-1}}{(\log t)^{s-2+o(1)}} \qquad (t\longrightarrow\infty).$

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