A polynomial improvement for the odd cycle-complete Ramsey numbers

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 $\varepsilon_{\ell} > 0$.