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Half a century ago, solving a conjecture of Erd\\H os, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all $m,s\\ge 1$. During the years very little progress in the general case was made.\n  In the present paper we state a general conjecture concerning the value of $e(sm-l,m)$ for $1<l<s$ and prove its validity for $s>s_0(l,m).$ For $l=2$ we determine the value of $e(sm-2,m)$ for all $s\\ge 5.$\n  ","authors_text":"Andrey Kupavskii, Peter Frankl","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-20T20:55:08Z","title":"Families with no $s$ pairwise disjoint sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06122","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:00cbafdc368d905f3e32186cb9503ec25eb5ec484821f2492fab044f06829850","target":"record","created_at":"2026-05-18T00:29:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"082e63be6779e6837e43eec9cfed1ab326f2eceb294046b2f306bde67f2620e5","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-20T20:55:08Z","title_canon_sha256":"7fcb4727748429b7099c629b499cd169924806b8a86b1083ebac48eda62f932c"},"schema_version":"1.0","source":{"id":"1607.06122","kind":"arxiv","version":2}},"canonical_sha256":"ad48ba42007a4e9931e5569bcf3439e596c6b7e779f106909807eae582fb3fff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad48ba42007a4e9931e5569bcf3439e596c6b7e779f106909807eae582fb3fff","first_computed_at":"2026-05-18T00:29:15.582628Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:15.582628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AMWkbtiaRCXThvs+rFr9pIPTyfQH5lsrzZz3Ddu5UwIgcf6RFUoNZdp92T79uEOhgoMmQaIvzjMRjoxjeh/JCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:15.583104Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.06122","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:00cbafdc368d905f3e32186cb9503ec25eb5ec484821f2492fab044f06829850","sha256:4c6bf5a708716812858d4355b9a2c153b572783a29471a9c4e87a98ac07c5b06"],"state_sha256":"33ac22c88328c994c93e67accc7f625d87d6acdba1ba9689e5b16d3afcfb5d77"}