Let $G$ be a simple graph of order $n$. If $G$ is triangle-free, then we know that there is a bipartite graph of the same order and the same size. So $G$ has size less than $n^2/4$.
Now if it has triangles, it might have more than $n^2/4$ edges. But then if we think of the graph as a smallest union of triangles and edges, then I guess that this union involves less than $n^2/4$ triangles or edges. I tried with several examples and my guess holds for them.
Is this a known result, or is it just false?