TITLE: How big are transitively reduced graphs?
QUESTION [9 upvotes]: What is the largest transitively reduced acyclic connected graph on $n$ vertices, for every $n$?
How many edges $e$ does it have?  How does $e$ grow as a function of $n$?  Faster than $n^2/4$?

REPLY [6 votes]: Since the transitively reduced graph $G$ is acyclic, there is a corresponding undirected simple graph $G'$, which can be formed by ignoring the direction of the edge. The number of edges in $G$ is the same as the number of edges in $G'$.
For $G$ to be transitively reduced, a necessary condition is that $G'$ must not contain any triangles.
It is well known (Turan's/Mantel's theorem) that any simple undirected graph on $n$ vertices and more than $n^2/4$ edges has a triangle.
It is also known that the undirected graph on $n$ vertices with maximum edges and no triangles is the complete bipartite graph $K_{[n/2],[n/2]}$ (for instance check out: http://books.google.com/books?id=SbZKSZ-1qrwC&pg=PA28, exercise 4)
From your comments, it looks like you found a $G$ whose corresponding $G'$ is $K_{[n/2],[n/2]}$ and so that proves it.