{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VUHPL2CLI6XS5UNCOHMMXPA347","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9afb4e47796125c312e936ecdfc7cb3ecd2241c3c2f04d864f454f56262fe4a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-07T20:08:02Z","title_canon_sha256":"a172b330ec575a39bab3b53e8ac80fb0debba69b5aa3b2fc2f49e6807d8c3177"},"schema_version":"1.0","source":{"id":"1604.02160","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.02160","created_at":"2026-05-18T00:15:02Z"},{"alias_kind":"arxiv_version","alias_value":"1604.02160v4","created_at":"2026-05-18T00:15:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02160","created_at":"2026-05-18T00:15:02Z"},{"alias_kind":"pith_short_12","alias_value":"VUHPL2CLI6XS","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VUHPL2CLI6XS5UNC","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VUHPL2CL","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:403213e090ff27c5c5b58afc87139118d9441f3051655f2e94ff9bc1c00fe72a","target":"graph","created_at":"2026-05-18T00:15:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Erd\\H{o}s-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a maximum-sized family.\n  In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. Our approach is rather general, and allows the leveraging of a wide variety of ","authors_text":"David Ellis, Nathan Keller, Noam Lifshitz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-07T20:08:02Z","title":"Stability versions of Erd\\H{o}s-Ko-Rado type theorems, via isoperimetry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02160","kind":"arxiv","version":4},"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":"sha256:8d6d719b4fdf05d69cc1bc5b67e17cbe2b62335d200f336ab5a2eaad2866b752","target":"record","created_at":"2026-05-18T00:15:02Z","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":"9afb4e47796125c312e936ecdfc7cb3ecd2241c3c2f04d864f454f56262fe4a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-07T20:08:02Z","title_canon_sha256":"a172b330ec575a39bab3b53e8ac80fb0debba69b5aa3b2fc2f49e6807d8c3177"},"schema_version":"1.0","source":{"id":"1604.02160","kind":"arxiv","version":4}},"canonical_sha256":"ad0ef5e84b47af2ed1a271d8cbbc1be7d79d07427343ee6903ba16e8b7eedf08","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad0ef5e84b47af2ed1a271d8cbbc1be7d79d07427343ee6903ba16e8b7eedf08","first_computed_at":"2026-05-18T00:15:02.472635Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:15:02.472635Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kMqtR0xzVX4xirx/2vB2KqIr7e8vluwLWm9mww2/tRpfdPZRGGfwu5ATCIEfmLA38mb4E3PiVCxDuYMTIe7FAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:15:02.473206Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.02160","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8d6d719b4fdf05d69cc1bc5b67e17cbe2b62335d200f336ab5a2eaad2866b752","sha256:403213e090ff27c5c5b58afc87139118d9441f3051655f2e94ff9bc1c00fe72a"],"state_sha256":"fd6f9f857070548337fea8bad8cfe0e883af175d5aed98ec34698aa7ea67cf6a"}