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Assuming that neither $\\mathscr{A}$ nor $\\mathscr{B}$ is empty, we prove several best possible bounds. In particular, we show that $|\\mathscr{A}|+|\\mathscr{B}|\\le 1+\\sum\\limits_{0\\le i\\le s}{{n}\\choose{i}}$. Supposing $n\\ge 2s$ and $\\mathscr{A},\\mathscr{B}$ are antichains, we show that $|\\mathscr{A}|+|\\mathscr{B}|\\le {{n}\\choose{1}}+{{n}\\choose{s-1}}$ unless $\\mat"},"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":"2006.12602","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-22T20:33:51Z","cross_cats_sorted":[],"title_canon_sha256":"8dc43ceb0dc7045bbb149f5e4b1264d27b25b12ced6f7ae1c30d0646bc473784","abstract_canon_sha256":"4c305f36fdda40a1aca39914d890a03f6a7dfa99432ccfa0ae2531a04deddf00"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:28:49.877042Z","signature_b64":"Evr8XPqoiBnbaoWYWk92ZDh22ymWf3lts18p2VTodEUYHAFRGFAZdiVYKb3QyCXzpb0rEkcVAXVNZwQGCCEgCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ded8e9b2850280e55998b5c32ec0da884f9cf5d9bc8ebd6a51aa5aab8006b79c","last_reissued_at":"2026-07-05T02:28:49.876680Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:28:49.876680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Analogues of Katona's and Milner's Theorems for two families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Peter Frankl, Willie Wong H.W","submitted_at":"2020-06-22T20:33:51Z","abstract_excerpt":"Let $n>s>0$ be integers, $X$ an $n$-element set and $\\mathscr{A}, \\mathscr{B}\\subset 2^X$ two families. 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