We consider the class $C$ of directed simple (no multiple edges) graphs having the property that every vertex is reachable by a directed path from every other vertex.

Given an integer $k$, what is the maximal possible number of (directed) edges in a graph of $C$ with $n$ vertices such that there are no directed cycles of length $\leq k$?

For $k=2$ this means simply that the existence of an edge from $v$ to $w$ forbids the existence of an edge from $w$ to $v$ and one can thus choose arbitrary orientions (giving rise to a graph in $C$) on the edges of the complete unoriented graph.

For $k=3$, one has also to forbid oriented triangles which is not possible by orienting all edges of a complete graph on $n\geq 3$ vertices such that the result is in $C$.

On the other hand, there are of course no triangles by choosing arbitrary orientations (giving rise to an element in $C$) of a complete bipartite graph. There are thus such graphs having roughly $n^2/4$ directed edges.

There should be better upper and lower bounds.

Motivation: G. Higman (A finitely generated infinite simple group. J. London Math. Soc. 26, (1951). 61--64) constructed finitely generated infinite simple groups by considering quotients of the finitely presented group $$\langle g_1,\dots,g_n|g_{i-1}^{-1}g_ig_{i-1}=g_i^2\rangle$$ where indices are modulo $n$.

This group is trivial for $n=2,3$ and infinite for $n\geq 4$. Given a directed graph, one can consider the corresponding group-presentation with generators corresponding to vertices and directed edges corresponding to relations $a^{-1}ba=b^2$. The triviality of the group constructed by Higman associated to $n=2,3$ implies that we want to avoid oriented cycles of length $\leq 3$ when searching for interesting examples.