Rainbow Directed Cycles of Order 4 in the Complete Biorientation of $$K_n$$

Let $$\overleftrightarrow {K_n}$$ be the complete biorientation of $$K_n$$ . Given a coloring of the arcs of $$\overleftrightarrow {K_n}$$ , a subdigraph D of $$\overleftrightarrow {K_n}$$ will be called rainbow if no two arcs of D receive the same color. For an integer n, let $$\overrightarrow{ar}(n, \overrightarrow{C_4})$$ be the minimum integer k such that every coloring of the arcs of $$\overleftrightarrow {K_n}$$ with k colors produces a rainbow copy of the directed cycle of order 4. In this paper, we show that for $$n\ge 9$$ , $$ \overrightarrow{ar}(n, \overrightarrow{C_4})=\tau (n,4)+2$$ , where $$\tau (n,4)$$ is the Turán number of $$K_4$$ , and characterize the extremal colorings for $$n\ge 12$$ (also bounds for the small cases are presented). Using these results it is proved that for $$n\ge 9$$ , $$ex(n, \overrightarrow{P_4}) = \tau (n,4)$$ where $$ex(n, \overrightarrow{P_4})$$ is the maximum size of a simple digraph free of directed paths of order 4 (the Turán number of $$\overrightarrow{P_4}$$ ).