{"paper":{"title":"Cover-free families on graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For any simple graph G, the minimum t for a G-Sperner family equals t(1, χ(G)).","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lucia Moura, Prangya Parida","submitted_at":"2026-05-12T18:17:08Z","abstract_excerpt":"A family of subsets of a $t$-set is a \\emph{$d$-cover-free family} or $d$-CFF if no subset in the family is contained in the union of any $d$ other subsets. Let $t(d, n)$ denote the minimum $t$ for which there exists a $d$-CFF on a $t$-set with $n$ subsets. Since a $1$-CFF is the same as a Sperner family, using Sperner's theorem, we get $t(1, n) \\sim \\log_{2}(n)$ as $n$ grows. Erd\\\"os, Frankl, and F\\\"uredi (JCTA, 1982) proved that $3.106\\log_{2}(n) < t(2,n) < 5.512\\log_{2}(n)$. This paper focuses on generalizing $1$-CFF and $2$-CFF using a graph $G$ where vertices correspond to subsets in the "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove t_s(G) = t(1, χ(G)) for any simple graph G.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The graph is simple with no isolated vertices when applying the trivial bound t(1,n) ≤ t(G) ≤ t(2,n); constructions for specific graphs assume standard combinatorial encodings like mixed-radix representations exist without hidden overlaps.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For any graph G, the minimum universe size for G-Sperner families equals the chromatic number of G, while G-cover-free families on paths and cycles satisfy log2(n) ≤ t ≤ 1.893 log2(n) + O(1).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For any simple graph G, the minimum t for a G-Sperner family equals t(1, χ(G)).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"32b59b53e94e92bb10bc17a82bb88775969c0795085ff59e4c5525bf9456ea64"},"source":{"id":"2605.12634","kind":"arxiv","version":1},"verdict":{"id":"060a1277-0dff-4f8e-b3d2-fb8ae21438c7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:25:35.930574Z","strongest_claim":"We prove t_s(G) = t(1, χ(G)) for any simple graph G.","one_line_summary":"For any graph G, the minimum universe size for G-Sperner families equals the chromatic number of G, while G-cover-free families on paths and cycles satisfy log2(n) ≤ t ≤ 1.893 log2(n) + O(1).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The graph is simple with no isolated vertices when applying the trivial bound t(1,n) ≤ t(G) ≤ t(2,n); constructions for specific graphs assume standard combinatorial encodings like mixed-radix representations exist without hidden overlaps.","pith_extraction_headline":"For any simple graph G, the minimum t for a G-Sperner family equals t(1, χ(G))."},"references":{"count":29,"sample":[{"doi":"","year":2015,"title":"Linear time constructions of some-restriction problems","work_id":"01e2c079-9164-4151-bb16-590a28ddc0b0","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"An efficient algorithm for group testing with runlength constraints.Discrete Applied Mathematics, 360:181–187, 2025","work_id":"a318147e-dd5d-4b9f-9021-4e2b5833ef9a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1987,"title":"A decomposition theorem for partially ordered sets.Classic papers in combinatorics, pages 139–144, 1987","work_id":"2ca86912-68f2-4bb1-bfe2-5aa1fd2674f0","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"Dingzhu Du and Frank Hwang.Combinatorial group testing and its applications, volume 12. World Scientific, 2000","work_id":"3d75fe96-f050-44f6-a726-f1b63dda69bc","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1982,"title":"Bounds on the length of disjunctive codes.Problemy Peredachi Informatsii, 18(3):7–13, 1982","work_id":"98b0c126-69f4-4fa4-a8bd-33958efda8e7","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":29,"snapshot_sha256":"e7602d2a41c277684dde2fc0bac205ff8c5d196a9265237656e81b57b79199f6","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"}