A polynomial improvement for the odd cycle-complete Ramsey numbers

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 $k\to \infty$, for some ${\epsilon }_{\ell }>0$.