A polynomial improvement for the odd cycle-complete Ramsey numbers

We give a polynomial improvement to the cycle-complete Ramsey numbers $$r(C_{\mathrm{\ell }},K_k)\ge k^{1+1\mathrm{/}(\mathrm{\ell }-2)+{\epsilon }_{\mathrm{\ell }}+o(1)},$$ for all fixed odd $\mathrm{\ell }>7$ with $k\to \mathrm{\infty }$, for some ${\epsilon }_{\mathrm{\ell }}>0$.