Cheers I came across the following: True or False question: Is there a number $n_0$ s.t. every graph with $n \geq n_0$ contains a $\lfloor n/2\rfloor $ clique or $\lfloor n/2\rfloor $-independent set?
I am thinking it's false, but I am also trying to prove it. Deducting it to Ramsey's numbers, I guess what I have to show is that $R(\lfloor n/2\rfloor,\lfloor n/2\rfloor) = n_0$, thus showing that every graph with $n_0$ or more vertices contains the $\lfloor n/2\rfloor$ clique or independent set (from the very definition of the Ramsey Numbers). Knowing that $R(k,m) > (k-1)(m-1) \Rightarrow R(\lfloor n/2\rfloor,\lfloor n/2\rfloor) > (\lfloor n/2\rfloor - 1)^2 \Rightarrow n_0 > (\lfloor n/2\rfloor - 1)^2 $ which of course doesn't hold for $n \to \infty$. I guess what I am trying to prove is that even I fix $n_0$, as $n$ gets bigger, we will reach a point where the Ramsey number will be bigger that $n_0$, and thus $n_0$ will not hold any more. Am I going in the right direction? Thanks