{"data":{"id":"10.5281/zenodo.22568303","type":"dois","attributes":{"doi":"10.5281/zenodo.22568303","prefix":"10.5281","suffix":"zenodo.22568303","identifiers":[{"identifier":"oai:zenodo.org:22568303","identifierType":"oai"}],"alternateIdentifiers":[{"alternateIdentifierType":"oai","alternateIdentifier":"oai:zenodo.org:22568303"}],"creators":[{"name":"Zeraoulia, Rafik","nameType":"Personal","givenName":"Rafik","familyName":"Zeraoulia","affiliation":["Université Djilali Bounaama Khemis Miliana"],"nameIdentifiers":[{"nameIdentifier":"0000-0002-5436-3320","nameIdentifierScheme":"ORCID"}]}],"titles":[{"title":"The Erdős–Sauer Clique Decomposition Problem for 3-Graphs: Verification through Nine Vertices and Local Packing Reductions"}],"publisher":"Zenodo","container":{},"publicationYear":2026,"subjects":[{"subject":"Erdős–Sauer problem clique decomposition 3-uniform hypergraphs hypergraph decomposition tetrahedron packing K_4^3 Turán numbers extremal hypergraph theory Steiner systems S(3,4,8) S(3,4,10) hypergraph packing hypergraph covering maximum packing combinatorial designs"}],"contributors":[],"dates":[{"date":"2026-09-07","dateType":"Issued"}],"language":"en","types":{"ris":"GEN","bibtex":"misc","citeproc":"article","schemaOrg":"CreativeWork","resourceType":"","resourceTypeGeneral":"Preprint"},"relatedIdentifiers":[{"relationType":"IsVersionOf","relatedIdentifier":"10.5281/zenodo.22566165","relatedIdentifierType":"DOI"}],"relatedItems":[],"sizes":[],"formats":[],"version":"1.1","rightsList":[{"rights":"Creative Commons Attribution 4.0 International","rightsUri":"https://creativecommons.org/licenses/by/4.0/legalcode","schemeUri":"https://spdx.org/licenses/","rightsIdentifier":"cc-by-4.0","rightsIdentifierScheme":"SPDX"},{"rights":"© 2026 Rafik Zeraoulia.   This work is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).   You are free to share and adapt this material for any purpose, even commercially, provided you give appropriate credit to the author and indicate if changes were made.   License details: https://creativecommons.org/licenses/by/4.0/","rightsUri":"http://rightsstatements.org/vocab/InC/1.0/"}],"descriptions":[{"description":"We study the r=3 case of the Erdős–Sauer clique decomposition problem and prove that its conjectured bound holds for every 3-graph on at most nine vertices. If G is a 3-graph, m=|E(G)|, and ν(G) is the maximum number of pairwise edge-disjoint tetrahedra K_4^3 in G, then the minimum number of pieces in a decomposition into single triples and tetrahedra is m-3ν(G). Hence the problem is equivalent to\n\nm-3ν(G) ≤ ex_3(n,K_4^3).\n\nThe finite result through n=9 is the first main contribution of the paper. The cases through n=7 follow from the near-Turán reduction together with the exact small Turán numbers, while n=8 and n=9 are proved analytically. The n=8 argument uses an S(3,4,8), and the n=9 argument uses an S(3,4,10) together with a collision estimate for missing triples under random relabelling. A recent working report independently records the packing identity and verifies the range through n=7; accordingly, the new finite advance claimed here is the analytic extension from seven to nine vertices.\n\nThe second contribution is a deterministic local theory around a maximum tetrahedron packing. We prove a leaf-rigidity lemma, a pair-dominant assignment criterion, and a good-orientation criterion on an actual-bridge graph, whose edges represent tetrahedra actually witnessed in G rather than mere support overlaps. We also prove an internal four-pack cover of cost at most 11, improved to 10 under a mild intersection condition, and formulate a boundary-credit criterion that converts this internal saving into a genuine 3-peeling step when the residual boundary has small cover number. These structural results identify concrete sufficient conditions for the conjectured inequality, but they do not yet force such a condition in every maximum packing. No proof of the general r=3 conjecture is claimed.","descriptionType":"Abstract"}],"geoLocations":[],"fundingReferences":[],"xml":"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","url":"https://zenodo.org/doi/10.5281/zenodo.22568303","contentUrl":null,"metadataVersion":0,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"api","isActive":true,"state":"findable","reason":null,"viewCount":0,"viewsOverTime":[],"downloadCount":0,"downloadsOverTime":[],"referenceCount":0,"citationCount":0,"citationsOverTime":[],"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":1,"created":"2026-09-07T01:20:29.000Z","registered":"2026-09-07T01:20:29.000Z","published":"2026","updated":"2026-09-07T01:20:29.000Z"},"relationships":{"client":{"data":{"id":"cern.zenodo","type":"clients"}},"provider":{"data":{"id":"cern","type":"providers"}},"media":{"data":{"id":"10.5281/zenodo.22568303","type":"media"}},"references":{"data":[]},"citations":{"data":[]},"parts":{"data":[]},"partOf":{"data":[]},"versions":{"data":[]},"versionOf":{"data":[{"id":"10.5281/zenodo.22566165","type":"dois"}]}}}}