Not sure how to proceed. I'm trying to prove that the following inequality is true. I know that $t_2 = 6$ and $t_3=17$ from the problem statement. The base case is obvious.
$t_{r+1} \leq (r+1) (t_r - 1) + 2$
($t_r$ is supposed to be the Ramsey number $R(3,3,3,\ldots,3)$ with $r$ $3$s).