  Let $R_k(H;K_m)$ be the smallest number $N$ such that every coloring of the
edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$
for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$.
In this short note, we study the lower bound for $R_k(H;K_m)$ when $H$ is $C_5$
or $C_7$, respectively. We show that
  \begin{equation*}
  R_{k}(C_5;K_m)=\Omega(m^{\frac{3k}{8}+1}/(\log{m})^{\frac{3k}{8}+1}),
  \end{equation*} and
  \begin{equation*}
  R_{k}(C_7;K_m)=\Omega(m^{\frac{2k}{9}+1}/(\log{m})^{\frac{2k}{9}+1}),
  \end{equation*} for fixed positive integer $k$ and $m\rightarrow\infty$.
These slightly improve the previously known lower bound
$R_{k}(C_{2\ell+1};K_m)=\Omega(m^{\frac{k}{2\ell-1}+1}/(\log
m)^{k+\frac{2k}{2\ell-1}})$ obtained by Alon and R\"{o}dl. The proof is based
on random block constructions and random blowups argument.
