A Polynomial Improvement for the Odd Cycle–Complete Ramsey Numbers

Abstract We give a polynomial improvement to the cycle-complete Ramsey numbers $$\begin{aligned} r(C_{\ell },K_k) \geqslant k^{1+\frac{1}{\ell -2} + \varepsilon _{\ell } + o(1)}, \end{aligned}$$ r ( C ℓ , K k ) ⩾ k 1 + 1 ℓ - 2 + ε ℓ + o ( 1 ) , for all fixed odd $$\ell > 7$$ ℓ > 7 with $$k \rightarrow \infty $$ k → ∞ , for some $$\varepsilon _{\ell } > 0$$ ε ℓ > 0 .