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arXiv · 2603.18565

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

Jianxi Liu
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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.

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Jianxi Liu. 2026-03-19. The typical structure of oriented graphs and digraphs with forbidden blow-up of transitive tournaments. https://arxiv.org/abs/2603.18565

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