The number $4/9$ is a non-jump for $3$-graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916009692102656 |
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| author | Liu, Xizhi Mubayi, Dhruv |
| author_facet | Liu, Xizhi Mubayi, Dhruv |
| contents | We prove that $4/9$ is a non-jump for $3$-uniform hypergraphs. Our construction perturbs the $ABB$ pattern by inserting, inside the $B$-part, the union of a high-cogirth pair of Steiner triple systems. This goes below the barrier for non-jumps obtainable by Shaw's finite-pattern formulation of the Frankl--Rödl method introduced in 1984. All results employing this approach use patterns where one of the parts has complete shadow. As the $ABB$ pattern is the smallest one with this property, the value $4/9$ is the natural barrier using this technique, and we conjecture that $4/9$ is the smallest non-jump for $3$-graphs. If our conjecture is true, this would answer (in a very strong form) an old question of Erd\Hos. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13567 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The number $4/9$ is a non-jump for $3$-graphs Liu, Xizhi Mubayi, Dhruv Combinatorics We prove that $4/9$ is a non-jump for $3$-uniform hypergraphs. Our construction perturbs the $ABB$ pattern by inserting, inside the $B$-part, the union of a high-cogirth pair of Steiner triple systems. This goes below the barrier for non-jumps obtainable by Shaw's finite-pattern formulation of the Frankl--Rödl method introduced in 1984. All results employing this approach use patterns where one of the parts has complete shadow. As the $ABB$ pattern is the smallest one with this property, the value $4/9$ is the natural barrier using this technique, and we conjecture that $4/9$ is the smallest non-jump for $3$-graphs. If our conjecture is true, this would answer (in a very strong form) an old question of Erd\Hos. |
| title | The number $4/9$ is a non-jump for $3$-graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.13567 |