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Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \\leq ex(n, Q_2)\\leq 2.283261N +o(N), $ where $N = \\binom{n}{\\lfloor n/2 \\rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at "},"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":"0912.5039","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-26T20:27:08Z","cross_cats_sorted":[],"title_canon_sha256":"c2999a66e079d03e5024a6e72ae205f5f310b3afa6631270f8f2bc4d901d5477","abstract_canon_sha256":"55624edcd1d8d0c69981d7bbb0ccfe3184739bbfeb7b5ca36b356f9c50267569"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:18.714652Z","signature_b64":"G7mtbb7qgMqhGnv5SXI1/ALCx4ufrGZPmlDVBSCYQ9+WUYYXP71eVenMOWJ19UvVcuJbXjhAr2d52+bwYFpfAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abd4c6630cf4434652c3cdf4b050d4102b3ab33978e715f909f4e17af165605f","last_reissued_at":"2026-05-18T01:14:18.714093Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:18.714093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$Q_2$-free families in the Boolean lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jacob Manske, Maria Axenovich, Ryan R. Martin","submitted_at":"2009-12-26T20:27:08Z","abstract_excerpt":"For a family $\\mathcal{F}$ of subsets of [n]=\\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \\leq ex(n, Q_2)\\leq 2.283261N +o(N), $ where $N = \\binom{n}{\\lfloor n/2 \\rfloor}$. 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