  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 $n\geq l\geq 3$ except for $n=l=3$.
Nikiforov proved this conjecture for $n\geq 4l+2$. 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 $n\geq 4\kappa+2$.
