{"paper":{"title":"The number $4/9$ is a non-jump for $3$-graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The number 4/9 is a non-jump for 3-uniform hypergraphs.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dhruv Mubayi, Xizhi Liu","submitted_at":"2026-05-13T14:05:39Z","abstract_excerpt":"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--R\\\"odl 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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that 4/9 is a non-jump for 3-uniform hypergraphs.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That inserting the union of a high-cogirth pair of Steiner triple systems into the B-part of the ABB pattern produces a valid construction achieving the non-jump property at 4/9.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"4/9 is a non-jump for 3-graphs via a perturbed ABB construction inserting high-cogirth pairs of Steiner triple systems.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The number 4/9 is a non-jump for 3-uniform hypergraphs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"568c421408d278550120626f082d1693f651551e0984597f862ff44c1084d1e9"},"source":{"id":"2605.13567","kind":"arxiv","version":1},"verdict":{"id":"88a06edb-b63c-42be-9160-d815158e0797","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:39:27.037216Z","strongest_claim":"We prove that 4/9 is a non-jump for 3-uniform hypergraphs.","one_line_summary":"4/9 is a non-jump for 3-graphs via a perturbed ABB construction inserting high-cogirth pairs of Steiner triple systems.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That inserting the union of a high-cogirth pair of Steiner triple systems into the B-part of the ABB pattern produces a valid construction achieving the non-jump property at 4/9.","pith_extraction_headline":"The number 4/9 is a non-jump for 3-uniform hypergraphs."},"references":{"count":35,"sample":[{"doi":"","year":2011,"title":"R. Baber and J. Talbot. Hypergraphs do jump.Combin. Probab. Comput., 20(2):161–171, 2011","work_id":"674a2c85-3f8d-47b7-a8de-26e729724123","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"T. Bohman and L. Warnke. Large girth approximate Steiner triple systems.J. Lond. Math. Soc. (2), 100(3):895–913, 2019","work_id":"03a9c2ec-8712-4700-bb21-481af78d55be","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1984,"title":"W. G. Brown and M. Simonovits. Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures.Discrete Math., 48(2–3):147–162, 1984","work_id":"2fd47c73-4b90-4370-9b90-e687f9949b94","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"M. Delcourt and L. Postle. Proof of the High Girth Existence Conjecture via refined absorption,","work_id":"c26a5107-8237-4fab-9777-cac4b70cdd29","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1964,"title":"P. Erdős. On extremal problems of graphs and generalized graphs.Israel J. Math., 2:183–190, 1964. 9","work_id":"8aed23f5-7048-4edf-8974-7677fa0dd660","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":35,"snapshot_sha256":"3e0a9827e01d4a2959896a7137cdc4e91c4b7adf4b817e0626e3af6741bec211","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}