Extremal graph theory

Question

Determine ex(n,2K2) for every n. (2K2 means a pair of vertex-disjoint edges, ex(n,H) = max{e(G): |G| = n is H-free})

I think the answer might be n+1 choose 2 but I am stuck on where to start.

Answer 1

We shall show that ex(n, 2K2) = n-1. For a construction where there are n-1 edges, consider the tree with 1 vertex connected to every other vertex, and every pair of 2 edges has 1 vertex in common, the first vertex.

Now, suppose that a graph with n vertices has at least n edges, we shall show that this graph has 2K2, i.e. a pair of vertex-disjoint edges. Suppose it does not, and that every pair of edges has a vertex in common. Consider 3 edges; taken pairwise they each have a vertex in common, hence all three edges have a vertex in common. Therefore, every 3 edges have a common vertex. It can be shown through induction that every k edges have a common vertex, 1