Turán theorems and convexity invariants for directed graphs

This paper is motivated by the desire to evaluate certain classical convexity invariants (specifically, the Helly and Radon numbers) in the context of transitive closure of arcs in the complete digraph. To do so, it is necessary to establish several new Turán type results for digraphs and characterize the associated extremal digraphs. In the case of the Radon number, we establish the following analogue for transitive closure in digraphs of Radon's classical convexity theorem [J. Radon, Mengen konvexer Körper, die einer gemeinsamen Punkt enthalten, Math. Ann. 83 (1921) 113–115]: in a complete digraph on n⩾7n⩾7 vertices with >n2/4>n2/4 arcs, it is possible to partition the arc set into two subsets whose transitive closures have an arc in common.