The Ramsey number of a long cycle and complete graphs

In this paper, we prove that the multicolored Ramsey number $R(G_1,\dots ,G_n,K_{n_1},\dots ,K_{n_r})$ is at least $(\gamma -1)(\kappa -1)+1$ for arbitrary connected graphs $G_1,\dots ,G_n$ and $n_1,\dots ,n_r\in \mathbb{N}$, where $\gamma =R(G_1,\dots ,G_n)$ and $\kappa =R(K_{n_1},\dots ,K_{n_r})$. Erd\H os at al. conjectured that $R(C_n,K_l)=(n-1)(l-1)+1$ for every $n\ge l\ge 3$ except for $n=l=3$. Nikiforov proved this conjecture for $n\ge 4l+2$. Using the above bound, we derive the following generalization of this result. $R(C_n,K_{n_1},\dots ,K_{n_r})=(n-1)(\kappa -1)+1$, where $\kappa =R(K_{n_1},\dots ,K_{n_r})$ and $n\ge 4\kappa +2$.