For every fixed odd ℓ > 7 the Ramsey number satisfies r(C_ℓ, K_k) ≥ k^{1 + 1/(ℓ-2) + ε_ℓ + o(1)} as k tends to infinity, for some positive ε_ℓ depending only on ℓ.
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A polynomial improvement for the odd cycle-complete Ramsey numbers
For every fixed odd ℓ > 7 the Ramsey number satisfies r(C_ℓ, K_k) ≥ k^{1 + 1/(ℓ-2) + ε_ℓ + o(1)} as k tends to infinity, for some positive ε_ℓ depending only on ℓ.