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For S = Z^n, this number was introduced by Aliev et al. [2014] who gave an explicit bound showing that c(Z^n,k) = O(k) holds for every fixed n. Recently, Chestnut et al. [2015] improved this to c(Z^n,k) = O(k (log log k)(log k)^{-1/3} ) and provided the lower bound c(Z^n,k) = Omega(k^{(n-1)/(n+1)}).\n  We provide a combinatorial description of c(S,k) in terms of polytopes w"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1602.07839","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-25T08:18:11Z","cross_cats_sorted":["math.MG","math.OC"],"title_canon_sha256":"e517895f028fc3e796098e2a1954eddc23ef94e2141a6b3dbfda6c2f74c57c96","abstract_canon_sha256":"3c7a68368708d322b6d78139f62bfbfb4cca7ced2f369be46312570464139c53"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:58.451386Z","signature_b64":"cfo6E7vjQnIj2soruybwJErOEs1ToJ0Q4nonx3J7fz38VBt4+Af0Bd4h5MZgReFgKOMDuFduH/asC5Ra6wdZAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b96592a371027a800b3239a2930c50d854912d111b67051dd2f6458ac437017","last_reissued_at":"2026-05-18T01:19:58.450780Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:58.450780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tight bounds on discrete quantitative Helly numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.OC"],"primary_cat":"math.CO","authors_text":"Bernardo Gonz\\'alez Merino, Gennadiy Averkov, Ingo Paschke, Matthias Henze, Stefan Weltge","submitted_at":"2016-02-25T08:18:11Z","abstract_excerpt":"Given a subset S of R^n, let c(S,k) be the smallest number t such that whenever finitely many convex sets have exactly k common points in S, there exist at most t of these sets that already have exactly k common points in S. 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