I previously asked a question on the upper bound on multicolor Ramsey Number. I was able to solve that with a hint! I'm currently stuck on finding the lower bound for the following Ramsey Number. First, define $R(\underbrace{3, 3, \dots, 3}_{2n}) = R_{2n}(3)$.
I need to show that $$R_{2n}(3) > 5^n$$
Any advice or hint on how to proceed with this? I've come across several inequalities involving upper bounds of Ramsey Numbers. I'm a little confused on how to come up with a lower bound in the first place, and use that to answer the question.