A Note on Triangle-Free Graphs
There is a natural, if imprecise, notion that the requirement of trianglefreeness on a graph G forces it into a bipartite-like form. In an extreme case, if G has n vertices and edges then Turán’s Theorem gives that it must be the bipartite graph Our concern here is what happens when e is smaller, must G still exhibit bipartite-like behavior. Very roughly, our answer is: Yes, if e ≫ n 3/2, No otherwise.
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