I am self studying studying an applied combinatorics course and trying to prove that:
$$R(\underbrace{3, 3, \cdots, 3}_{n}) = R_n(3) \leq 1 + n!(1 + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!})$$
I tried using induction and there is equality for $n=2$, i.e. $R(3,3) = 6$, but I'm stuck with the inductive step. I'm also having a little trouble in conceptually understanding what Ramsey Numbers mean, and maybe that's why I'm stuck. Would really really appreciate any help. Thank you!
Edited: I was able to solve it following @kabenyuk's suggestion on using the reasoning from this post: Upper bound on multicolor ramsey number