extremal combinatorics

Jul 2009
193
52
Can anyone check if i'm correct with this question?

write down a graph H with chromatic number r but \(\displaystyle ex(n,H)>ex(n,K_{r})\) for all sufficiently large n.

So i need to find a graph with chromatic number r and contains \(\displaystyle K_{r}\) with an extra edge. So can't i just add a vertex and join it to 2 other vertices and we get a graph with chromatic number r (just place the added vertex in a class with another vertex it is not connected to) which does contain \(\displaystyle K_{r}\).