{"paper":{"title":"An Erd\\H{o}s-Ko-Rado theorem for binary codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Every maximum 3-wise intersecting family of binary words is a star.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chi Hoi Yip, Shamil Asgarli","submitted_at":"2026-04-15T04:57:17Z","abstract_excerpt":"We study intersecting families of words from the Erd\\H{o}s-Ko-Rado perspective. When the alphabet size is $2$, a maximum intersecting family is not necessarily a star. However, we prove that every maximum $3$-wise intersecting family is a star. We also present a new proof of the known result for alphabets of size at least $3$: maximum intersecting families of words are exactly the stars."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that every maximum 3-wise intersecting family is a star. We also present a new proof of the known result for alphabets of size at least 3: maximum intersecting families of words are exactly the stars.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The results hold for sufficiently large word length n (implicit in EKR-type statements); the paper does not specify the exact threshold in the abstract, so the precise range of n for which the statements are claimed is not visible from the provided text.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For binary words, every maximum 3-wise intersecting family is a star; for alphabets of size at least 3, maximum intersecting families of words are exactly the stars.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every maximum 3-wise intersecting family of binary words is a star.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6b4775340a40f01bd2d0fda241f6775c71625b9cb9752cb2e5bed6178788f70d"},"source":{"id":"2604.13475","kind":"arxiv","version":1},"verdict":{"id":"eff9b3c5-8d48-4fe5-acb4-742d633d1964","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T13:35:59.971028Z","strongest_claim":"We prove that every maximum 3-wise intersecting family is a star. We also present a new proof of the known result for alphabets of size at least 3: maximum intersecting families of words are exactly the stars.","one_line_summary":"For binary words, every maximum 3-wise intersecting family is a star; for alphabets of size at least 3, maximum intersecting families of words are exactly the stars.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The results hold for sufficiently large word length n (implicit in EKR-type statements); the paper does not specify the exact threshold in the abstract, so the precise range of n for which the statements are claimed is not visible from the provided text.","pith_extraction_headline":"Every maximum 3-wise intersecting family of binary words is a star."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.13475/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":9,"sample":[{"doi":"","year":1986,"title":"K. Engel and P. Frankl. An Erd ¨os-Ko-Rado theorem for integer sequences of given rank.European J. Combin., 7(3):215–220, 1986","work_id":"ee2c770f-4245-4e6c-93b7-80124335d444","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1961,"title":"P. Erd ˝os, C. Ko, and R. Rado. Intersection theorems for systems of finite sets.Quart. J. Math. Oxford Ser. (2), 12:313–320, 1961","work_id":"cfaec8ff-9901-4f55-9bd9-155e11d0b09b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1980,"title":"P. Frankl and Z. F ¨uredi. The Erd¨os-Ko-Rado theorem for integer sequences.SIAM J. Algebraic Discrete Methods, 1(4):376–381, 1980","work_id":"d3eb43ab-6479-4006-a1f3-3773bb978486","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1999,"title":"P. Frankl and N. Tokushige. The Erd ˝os–Ko–Rado theorem for integer sequences.Combinatorica, 19(1):55–63, 1999","work_id":"875249e7-a211-40ac-ac69-d62aceb61d5e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"C. Godsil and K. Meagher.Erd ˝os-Ko-Rado theorems: algebraic approaches, volume 149 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016","work_id":"c123477d-9bc3-4e62-83cc-eeb3062da033","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":9,"snapshot_sha256":"c86ca4f4945f0db8c7d55c19471086bfa70306eca5b24b2835b53f03df70ab75","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"}