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Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\\mathcal{F}\\subseteq\\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\\varepsilon>0$ such that for every $k$ and $|X| \\geq n_0(k)$ there exists a saturated $k$-Sperner system $\\mathcal{F}\\subs"},"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":"1402.5646","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-23T18:17:48Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"0ce266e592c02c1a0f0a032877a2ba47277172f0a9a44ad96ddc458ff8caa5b9","abstract_canon_sha256":"f2a9cd993c9ecadd724623bf91777d58a2064f6ea81e605cbafd191ada441ff3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:47.409006Z","signature_b64":"+oR0WWfHt6B0H5T0BfJV0MdjdhmKk2p8ahv/PQVSMFb/oC0te/F+nhgUF/DrYip1paLplvRXIV0FFNvvZFr+Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ed72dcdf8452df53ba5bd6b97dc9ea62e7c11896ab7837cd2ad42f3fa01e453","last_reissued_at":"2026-05-18T02:44:47.408371Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:47.408371Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Saturated $k$-Sperner Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Alex Scott, Jonathan A. 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