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Define the distance of these pairs by $d(\\{A_1,A_2\\} ,\\{B_1,B_2\\})=\\min \\{|A_1-B_1|+|A_2-B_2|, |A_1-B_2|+|A_2-B_1|\\} $. This is the minimum number of elements of $A_1\\cup A_2$ one has to move to obtain the other pair $\\{B_1,B_2\\}$. Let $C(n,k,d)$ be the maximum size of a family of pairs of disjoint subsets, such that the distance of any two pairs "},"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":"1403.3847","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-03-15T20:03:44Z","cross_cats_sorted":[],"title_canon_sha256":"c197926a26feb0152e0fbb1aa1f6c78ab618ce1ebbaec445fe7eba6d1435a4eb","abstract_canon_sha256":"011ea478ed2b924d93bd11a1d7a6999e29837168e0c424643b8c901058f7bc3c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:56.249953Z","signature_b64":"F/TGU+3wz6Dd54IoNPcYFscMLd9uMz09m4d/oA9/saLK52F+ouB57I35myz96jUdVUCA4YLGa2oDZUKsf1TJDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abe251ca054e323c7d4f533dca9f4b0df49873f31baf9334a964a2c0de933248","last_reissued_at":"2026-05-18T02:25:56.249565Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:56.249565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A coding problem for pairs of subsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bela Bollobas, G. O. H. Katona, Ida Kantor, Imre Leader, Zoltan Furedi","submitted_at":"2014-03-15T20:03:44Z","abstract_excerpt":"Let $X$ be an $n$--element finite set, $0<k\\leq n/2$ an integer. Suppose that $\\{A_1,A_2\\} $ and $\\{B_1,B_2\\} $ are pairs of disjoint $k$-element subsets of $X$ (that is, $|A_1|=|A_2|=|B_1|=|B_2|=k$, $A_1\\cap A_2=\\emptyset$, $B_1\\cap B_2=\\emptyset$). Define the distance of these pairs by $d(\\{A_1,A_2\\} ,\\{B_1,B_2\\})=\\min \\{|A_1-B_1|+|A_2-B_2|, |A_1-B_2|+|A_2-B_1|\\} $. This is the minimum number of elements of $A_1\\cup A_2$ one has to move to obtain the other pair $\\{B_1,B_2\\}$. 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