Characterization of Turánable oriented graphs via D_r

Determine whether every oriented graph H is Turánable if and only if there exists an integer r ≥ 1 such that H is a subgraph of D_r, where D_r is the tournament obtained from the r-fold blow-up of the directed 3-cycle C_3 by replacing each of the three independent parts with transitive tournaments.

Background

The authors define an oriented graph H to be Turánable if there exists a constant d ∈ (0,1/2] such that every sufficiently large n-vertex oriented graph G with minimum semi-degree at least d(n−1) must contain H. For tournaments, Bollobás and Hӓggkvist proved that a tournament T is Turánable if and only if T is a subgraph of D_r for some r.

The conjecture seeks to extend this characterization from tournaments to all oriented graphs. A positive resolution would unify Turán-type existence thresholds across the full class of oriented graphs and, combined with bounds for D_r, would yield general upper bounds for minimum semi-degree thresholds needed to force copies of many oriented graphs.