I want to show that $R_{k}(s) \leq 4^{s^{k-1}}$
(Notation: $R_{k}(s)$ = The least n such that when colouring $K_{n}$ with $k$-colours, we can find a monochromatic $K_{s}$)
From the proof that $R_{k}(s)$ exists, by induction, we use the induction hypothesis to say we can find $m$ such that if $K_{m}$ $(k-1)$-coloured then there exists a monochromatic $K_{s}$, then we let $n=R(s,m)$ and this $n$ works for $k$ colours (viewing the $(k-1)$ colours as a second colour and then using the induction hypothesis).
Now for the bound I've tried an induction using the proof above but can't seem to get to the end. Here is what I've tried:
$R_{k}(s) \leq n = R(s,m)$ where $m= R_{k-1}(s)$
$R_{k-1}(s) \leq 4^{s^{k-2}}$ by the induction hypothesis so,
$R_{k}(s) \leq R(s,4^{s^{k-2}})$
Now, a corollary of Ramsey's Theorem for 2 colours is $R(s,t) \leq 2^{s+t}$ and so,
$R_{k}(s) \leq 2^{s+4^{s^{k-2}}}$
From here I cannot get out the induction step to show the result. Am I on the right track? Is there an easier way to prove such a simple inequality? Although on my problem sheet it wants me to see the bound from the proof of Ramsey for $k$-colours.