{"paper":{"title":"Intersecting families, signed sets, and injection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carl Feghali","submitted_at":"2019-12-21T19:32:33Z","abstract_excerpt":"Let $k, r, n \\geq 1$ be integers, and let $\\S_{n, k, r}$ be the family of $r$-signed $k$-sets on $[n] = \\{1, \\dots, n\\}$ given by $$ \\mathcal{S}_{n, k, r} = \\Big\\{\\{(x_1, a_1), \\dots, (x_k, a_k)\\}: \\{x_1, \\dots, x_k\\} \\in \\binom{[n]}{k}, a_1, \\dots, a_k \\in [r] \\Big\\}. $$ A family $\\mathcal{A} \\subseteq \\S_{n, k, r}$ is \\emph{intersecting} if $A, B \\in \\mathcal{A}$ implies $A \\cap B \\not= \\emptyset$. A well-known result (first stated by Meyer and proved using different methods by Deza and Frankl, and Bollob\\'as and Leader) states that if $\\mathcal{A} \\subseteq \\mathcal{S}_{n, k, r}$ is interse"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.10324","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1912.10324/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}