triangle free and connected graph Let G be a connected and triangle-free graph with 82 vertices,
Prove that alpha(G)>=10
Where alpha(G) is the maximal independent set of G.
I've shown that if the maximal independent set is on 9 vertices , 
Then any other vertex we add to this set, it will be connected to at least one of them.
Then if we part the 73 vertices left to which one they're connected (if it is two different vertices choose one randomally)
We get one vertex of the independent set connected to atleast 9. If more then 9 we can find a triangle,
If 9 they create a new anti clique of 9 vertices. I don't know how to continue from here.
Thank you.

I just wanted to point out that we don't actually need $83$ vertices, any number of vertices greater than $R(10,3)$ will do, and in particular we have the upper bound for $R(10,3)$ of 42.
