Sharpness for orientations with orientation-invariant decomposition family

Prove that, for every directed graph H whose underlying graph satisfies M(H)=\overrightarrow{M(H)}, the equality D(n,H)=2^{ex(n,H)} holds for sufficiently large n.

Background

The paper introduces directed and undirected decomposition families to control the structure of H-free orientations. The condition M(H)=\overrightarrow{M(H)} means, in the paper’s notation, that every orientation of every graph in the relevant undirected decomposition family is represented in the directed decomposition family. Motivated by the exact results for the 2-fans and the anti-directed 3-fan, the authors propose this condition as a general sufficient criterion for equality.