The expansion property for classes of finite structures is a well studied Ramsey property for homogeneous structures. Recently, a quantitative version of this property was introduced to answer questions related to amenability and unique ergodicity of automorphism groups of homogeneous structures. A typical way to check this property involves fine estimates and the probabilistic method.

We introduce an even stronger expansion property that is purely combinatorial, while not being so strong as to be impossible. We will then classify which completely n-partite directed graphs have this property. Remarkably, the property is able to isolate the geometry of completely n-partite directed graphs.

This provides a step in the right direction towards the goal of showing that the semigeneric digraph has a uniquely ergodic automorphism group (which is still open).