Ramsey number $R(3,n) \ge 2(n-1)$

Question

I want to prove that $R(3,n) \ge 2(n-1)$ but I am not sure how to do that since using induction does not seem to work. Could anyone give me an idea? Thank you!

Answer 1

To show that the Ramsey number $R(a,b)$ is bounded below by $N$, you merely need to provide a construction of a complete graph with $N$ vertices that doesn't have a chromatic $a-$gon colored with $r_a$ or $b-$gon colored with $r_b$.

For $R(3, n) \geq 2(n-1)$, this is easy to do:

Hint: Consider $K_{n-1} \cup K_{n-1}$.