{"paper":{"title":"A counterexample to conjecture 18.5 in \"Geometric Etudes in Combinatorial Mathematics\", second edition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tobias Muller","submitted_at":"2011-12-31T13:04:52Z","abstract_excerpt":"A collection of sets $\\Fscr$ has the $(p,q)$-property if out of every $p$ elements of $\\Fscr$ there are $q$ that have a point in common. A transversal of a collection of sets $\\Fscr$ is a set $A$ that intersects every member of $\\Fscr$. Gr\\\"unbaum conjectured that every family $\\Fscr$ of closed, convex sets in the plane with the $(4,3)$-property and at least two elements that are compact has a transversal of bounded cardinality. Here we construct a counterexample to his conjecture. On the positive side, we also show that if such a collection $\\Fscr$ contains two {\\em disjoint} compacta then th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0254","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":""},"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"}