I am solving the following intermediate question to a problem about Ramsey Theorem.
Let $r_n=R(p_1,\ldots,p_n)$ with $p_i=3$ for each $i \in [n]$.
Given that $r_2=R(3,3)=6$ use $r_n \leq n (r_{n-1}-1)+2$ to show that $r_n\leq \lfloor n ! e \rfloor +1 $.
MY ATTEMPT
I show this property using induction.
(BASE CASE) $6=r_2\leq \lfloor 2!e \rfloor+1=\lfloor 5.43\ldots\rfloor+1$
(INDUCTIVE CASE) Let $n=k+1$. Then
$r_k\leq (k+1) (r_k-1)+2$
$\leq (k+1)(\lfloor k!e\rfloor)+2 $ {using IH}
$= ((k+1)(\lfloor k!e\rfloor)+1)+1 $
$= \lfloor k+1!e\rfloor+1$ {this seems to work by plugging values in a calculator, but how about actually proving it ?}
Questions:
Is the last step of the equality correct?
Is there a simpler way/alternative route on how one can go about this.