{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:WZOSBXWUVR3SYCHBN45KKAQUIT","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":"1ef242bfe2787bf846b4b394a2f8c7fb2e9b388c0b00670d3992cc678e469a04","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-18T11:05:53Z","title_canon_sha256":"8a3f15f7eb1f985675b2b2d9e0c9aaca5f0a93e68720b9d29013e32f8458f0f3"},"schema_version":"1.0","source":{"id":"1604.05084","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.05084","created_at":"2026-05-18T00:51:07Z"},{"alias_kind":"arxiv_version","alias_value":"1604.05084v2","created_at":"2026-05-18T00:51:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.05084","created_at":"2026-05-18T00:51:07Z"},{"alias_kind":"pith_short_12","alias_value":"WZOSBXWUVR3S","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"WZOSBXWUVR3SYCHB","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"WZOSBXWU","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:f181fc1518058029d96dc61cd02274d283ed098fe1b52e7c3e2f197d78714a03","target":"graph","created_at":"2026-05-18T00:51:07Z","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":"The Johnson graph $J(n,m)$ has the $m$--subsets of $\\{1,2,\\ldots,n\\}$ as vertices and two subsets are adjacent in the graph if they share $m-1$ elements. Shapozenko asked about the isoperimetric function $\\mu_{n,m}(k)$ of Johnson graphs, that is, the cardinality of the smallest boundary of sets with $k$ vertices in $J(n,m)$ for each $1\\le k\\le {n\\choose m}$. We give an upper bound for $\\mu_{n,m}(k)$ and show that, for each given $k$ such that the solution to the Shadow Minimization Problem in the Boolean lattice is unique, and each sufficiently large $n$, the given upper bound is tight. We als","authors_text":"Llu\\'is Vena, Oriol Serra, V\\'ictor Diego","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-18T11:05:53Z","title":"On a problem by Shapozenko on Johnson graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05084","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":"sha256:2c74255a9f457d19dc77d552266f3e578e153c02958005a6a23ad8acf8c4dfeb","target":"record","created_at":"2026-05-18T00:51:07Z","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":"1ef242bfe2787bf846b4b394a2f8c7fb2e9b388c0b00670d3992cc678e469a04","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-18T11:05:53Z","title_canon_sha256":"8a3f15f7eb1f985675b2b2d9e0c9aaca5f0a93e68720b9d29013e32f8458f0f3"},"schema_version":"1.0","source":{"id":"1604.05084","kind":"arxiv","version":2}},"canonical_sha256":"b65d20ded4ac772c08e16f3aa5021444c151ba072bafb779ebadfc7ae05d878b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b65d20ded4ac772c08e16f3aa5021444c151ba072bafb779ebadfc7ae05d878b","first_computed_at":"2026-05-18T00:51:07.313042Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:07.313042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KIqDGFUWwdOALenhMVWGspM9tZtHTg544+pl8/7V8w6tANQ44Imaq0Bv3fkBHXEMavSjuWUvvaSosJIFhB7iDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:07.313571Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.05084","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c74255a9f457d19dc77d552266f3e578e153c02958005a6a23ad8acf8c4dfeb","sha256:f181fc1518058029d96dc61cd02274d283ed098fe1b52e7c3e2f197d78714a03"],"state_sha256":"c85033aa4f22858327f913780060b29a03b2d7f9c61516d06341012ec358b6a0"}