The typical structure of oriented graphs and digraphs with forbidden blow-up of transitive tournaments
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| Format: | Preprint |
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2026
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| _version_ | 1866917371234484224 |
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| author | Liu, Jianxi |
| author_facet | Liu, Jianxi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_18565 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The typical structure of oriented graphs and digraphs with forbidden blow-up of transitive tournaments Liu, Jianxi Combinatorics 05C20, 05C35, 05C75, 05D40 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. |
| title | The typical structure of oriented graphs and digraphs with forbidden blow-up of transitive tournaments |
| topic | Combinatorics 05C20, 05C35, 05C75, 05D40 |
| url | https://arxiv.org/abs/2603.18565 |