Asymptotic upper-bound conjecture for the minimum size of k-dicritical oriented graphs

Let $o_k(n)$ be the minimum number of arcs in a $k$-dicritical oriented graph of order $n$. Suppose that, for each fixed $k$, there is a constant $c_k$ such that

$$o_k(n)=c_k n+o(n).$$

Asymptotic upper-bound conjecture.

$$\frac{c_k}{2k-3}\rightarrow 1\quad\text{when }k\rightarrow +\infty.$$

The paper establishes upper and lower bounds on $o_k(n)$ and proposes that the constructions giving the upper bound are nearly optimal, with the asymptotic growth rate approaching the upper bound as $k$ tends to infinity.