If $G$ is a graph with $n$ vertices and $\frac{nk}{2}$ edges, $k\ge -1,$ then $a(G)\ge \frac{n}{k+1}$. Why?


(Here $a(G)$ is the independence number).



Bounds on the independence number of a graph
by turan theorem, that is very simple: 


a(G)=w(G')≥n^2/(n^2-2(n(n-1)/2-m))=n^2/(2m+n)


