The logarithmic exponent of the off-diagonal Ramsey numbers r(s,t) for fixed s at least 5

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VibeMathed record: The logarithmic exponent of the off-diagonal Ramsey numbers r(s,t) for fixed s at least 5 evidence for this event

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The sharp logarithmic exponent of r(5,t) (companion, s = 5) evidence for this event

For every fixed $s\ge5$: there is $C_s$ such that for every $\varepsilon>0$ and all large $t$, $t^{s-1}/(\log t)^{s-2+\varepsilon}\le r(s,t)\le C_s t^{s-1}/(\log t)^{s-2}$, so $r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)}$ and the classical upper bound is sharp up to $(\log t)^{o(1)}$. The lower bound comes from a random ordered point-hyperplane flag graph in $\mathrm{PG}(s-1,q)$ (Bradac's construction) with a new entropy and compression analysis of its independent sets. It does NOT give $r(s,t)=\Theta(t^{s-1}/(\log t)^{s-2})$, and it does not cover $s=4$, where the gap between $t^3/(\log t)^4$ and $t^3/(\log t)^2$ remains. parent of this event

The logarithmic exponent of the off-diagonal Ramsey numbers r(s,t) for fixed s at least 5 parent of this event

The logarithmic exponent of the off-diagonal Ramsey numbers r(s,t) for fixed s at least 5 evidence for this event

Lean: OAI/Combinatorics/RamseyFive/Main.lean (SharpRamseyFive.main, s = 5) evidence for this event

Lean: OAI/Combinatorics/SharpRamsey/Main.lean (SharpLogRamsey.main, s >= 6) evidence for this event

Erdős Problem #986 evidence for this event