$r(k) := R(\underbrace{3,3,...,3}_k)$
(I.e. $r(k)$ is the minimum integer $n > 0$ such that every coloring of edges of $K_n$ in $k$ colors is guaranteed to produce a monochromatic triangle.) Show that for $k \ge 2$
$$r(k) \le k\big(r(k−1)−1\big) + 2$$
Can anyone help with this? Any hints are welcome! Thanks