## extremal combinatorics

Can anyone check if i'm correct with this question?

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

So i need to find a graph with chromatic number r and contains $K_{r}$ 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 $K_{r}$.