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

Abstract

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. For any integers r >= 2, t >= 1 and any real a in (3/2,2], we prove that almost all T_{r+1}^t-free oriented graphs and almost all T_{r+1}^t-free digraphs are r-partite. This extends the results of Kuhn, Osthus, Townsend and Zhao (2017) on forbidden transitive tournaments to their blow-ups, thereby confirming a generalised form of Cherlin's conjecture. Our proof combines the hypergraph container method, a weighted analogue of the Erdos-Stone theorem for digraphs, and a stability analysis for near-extremal T_{r+1}^t-free digraphs. The core of the proof is the interplay between the directed regularity lemma and an embedding lemma, which together provide a rigorous bridge from macroscopic extremal conditions to microscopic concrete structures.

Theorems & Definitions (22)

• Theorem 1.1: Main result – simplified form
• Theorem 1.2: Main result – precise form
• Lemma 2.1: Directed regularity lemma
• Lemma 2.2: Embedding lemma
• Theorem 2.3: Liu liu2026
• Lemma 2.4: Removal lemma
• Theorem 3.1
• proof
• Corollary 3.2
• Theorem 4.1: Stability theorem