Let $R_k(s)$ be the k-color Ramsey number (i.e., R_3(s)=R(s,s,s) and so on).
I want to show that $R_k(s)\leq k^{ks}$. My idea was to use induction on $s$ (for every $k$ fixed).
Assuming it is true for $s$, then I want to show that $R_k(s+1)\leq k^k\cdot k^{sk}$. I was looking at $k^k$ copies of $K_{k^{ks}}$ and trying to construct a monochromatic $K_{s+1}$ using that each one of these copies contains a monochromatic $K_s$.
Is this the correct approach or is there any easier way?