Proves r(s, k) ≥ Ω(k^{s-1} / (log k)^{2s-4}) for fixed s ≥ 3 and k → ∞, nearly matching the Erdős-Szekeres upper bound and improving the Spencer lower bound for s ≥ 5.
A polynomial improvement for the odd cycle-complete Ramsey numbers
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abstract
We give a polynomial improvement to the cycle-complete Ramsey numbers \[ r(C_{\ell},K_k) \geq k^{1+1/(\ell- 2) + \varepsilon_{\ell} + o(1)}, \] for all fixed odd $\ell > 7$ with $k \rightarrow \infty$, for some $\varepsilon_{\ell} > 0$.
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math.CO 1years
2026 1verdicts
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Off-diagonal Ramsey numbers
Proves r(s, k) ≥ Ω(k^{s-1} / (log k)^{2s-4}) for fixed s ≥ 3 and k → ∞, nearly matching the Erdős-Szekeres upper bound and improving the Spencer lower bound for s ≥ 5.