The typical structure of oriented graphs and digraphs with forbidden blow-up of transitive tournaments

For integers $r\ge 2$, $t\ge 1$ and a real number $a\in (3\mathrm{/}2,2]$, we study the typical structure of oriented graphs and digraphs that do not contain a blow-up $T_{r+1}^t$ of a transitive tournament. We prove that almost every $T_{r+1}^t$-free oriented graph on n vertices admits an r-partition $V_1\cup \cdots \cup V_r$ such that each induced subgraph $G[V_i]$ is $T_2^t$-free, and the same holds for almost every $T_{r+1}^t$-free digraph.Consequently, the number $f(n,T_{r+1}^t)$ of labelled $T_{r+1}^t$-free oriented graphs satisfies $f(n,T_{r+1}^t)=\mathrm{\mid }{\mathcal{P}}_{n,r,t}\mathrm{\mid }(1+o(1))$, where ${\mathcal{P}}_{n,r,t}$ is the family of oriented graphs admitting such an r-partition with each part $T_2^t$-free; an analogous statement holds for digraphs.When $t=1$ this recovers the result of K"uhn, Osthus, Townsend and Zhao (2017) that almost all $T_{r+1}$-free oriented graphs (resp. digraphs) are r-partite, thereby confirming a generalised form of Cherlin’s conjecture. Our proof combines the hypergraph container method, a weighted Erdős-Stone theorem, and a stability analysis for near-extremal $T_{r+1}^t$-free digraphs.