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Let $G$ be a graph with $n$ vertices and at least $n^2/4$ edges. Are there at least $2n^2/9$ edges of $G$ which are contained in a $C_5$?
Erdős [Er97d] stated that, under the same assumptions, there at least $2n^2/9$ edges of $G$ which are contained in some odd cycle. He wrote that a positive answer to this question would follow if we knew that $G$ must contain a triangle such that there at least $n/2-O(1)$ vertices joined to at least two vertices of the triangle.

Erdős and Faudree observed that every graph with $2n$ vertices and at least $n^2+1$ edges has a triangle whose vertices are joined to at least $n+2$ vertices.

See also the entry in the graphs problem collection.