{"paper":{"title":"On maximal S-free sets and the Helly number for the family of S-convex sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.OC","authors_text":"Gennadiy Averkov","submitted_at":"2011-09-28T12:00:54Z","abstract_excerpt":"We study two combinatorial parameters, which we denote by f(S) and h(S), associated to an arbitrary set S \\subseteq R^d, where d \\in N. In the nondegenerate situation, f(S) is the largest possible number of facets of a d-dimensional polyhedron L such that the interior of L is disjoint with S and L is inclusion-maximal with respect to this property. The parameter h(S) is the Helly number of the family of all sets that can be given as the intersection of S with a convex subset of R^d. We obtain the inequality f(S) \\le h(S) for an arbitrary S and the equality f(S)=h(S) for every discrete S. Furth"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.6179","kind":"arxiv","version":3},"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"}