{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:JE5PGAI4EHROZC5SEISZ5COKSZ","short_pith_number":"pith:JE5PGAI4","canonical_record":{"source":{"id":"1005.5622","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-31T08:51:53Z","cross_cats_sorted":[],"title_canon_sha256":"202c2fa38392265af49c76a02a307e49189bb9e50ca732ba6c038a467eafcd1b","abstract_canon_sha256":"1363801cf083cd0ca8d899a120226cdd24242f551781db3076c48645aa81727e"},"schema_version":"1.0"},"canonical_sha256":"493af3011c21e2ec8bb222259e89ca965360f79046edac834688027113a4fceb","source":{"kind":"arxiv","id":"1005.5622","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.5622","created_at":"2026-05-18T04:19:33Z"},{"alias_kind":"arxiv_version","alias_value":"1005.5622v3","created_at":"2026-05-18T04:19:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.5622","created_at":"2026-05-18T04:19:33Z"},{"alias_kind":"pith_short_12","alias_value":"JE5PGAI4EHRO","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"JE5PGAI4EHROZC5S","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"JE5PGAI4","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:JE5PGAI4EHROZC5SEISZ5COKSZ","target":"record","payload":{"canonical_record":{"source":{"id":"1005.5622","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-31T08:51:53Z","cross_cats_sorted":[],"title_canon_sha256":"202c2fa38392265af49c76a02a307e49189bb9e50ca732ba6c038a467eafcd1b","abstract_canon_sha256":"1363801cf083cd0ca8d899a120226cdd24242f551781db3076c48645aa81727e"},"schema_version":"1.0"},"canonical_sha256":"493af3011c21e2ec8bb222259e89ca965360f79046edac834688027113a4fceb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:33.511056Z","signature_b64":"cGlm8gkm+/AnzMmpElbvF3JMWbsAAl1FbVMFoE9CCWLGLS0cozLV2xaCGgXitj1OvadP87knaS5gTq7JZS9xDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"493af3011c21e2ec8bb222259e89ca965360f79046edac834688027113a4fceb","last_reissued_at":"2026-05-18T04:19:33.510659Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:33.510659Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1005.5622","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:19:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FyWeIXC43NgoYnrsv1Et1jEzmL3KW3MFrEOrP6ryBSG4wE85Z5xLqLnGjxMhcyBAeo6PM2nDzH3Y1KHO8J5LAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T04:57:53.988781Z"},"content_sha256":"2630642192cb0bb2b92a5edb0f9ca153e8c40240c932a344d8585a126cb433da","schema_version":"1.0","event_id":"sha256:2630642192cb0bb2b92a5edb0f9ca153e8c40240c932a344d8585a126cb433da"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:JE5PGAI4EHROZC5SEISZ5COKSZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the hypercompetition numbers of hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Boram Park, Yoshio Sano","submitted_at":"2010-05-31T08:51:53Z","abstract_excerpt":"The competition hypergraph $C{\\cH}(D)$ of a digraph $D$ is the hypergraph such that the vertex set is the same as $D$ and $e \\subseteq V(D)$ is a hyperedge if and only if $e$ contains at least 2 vertices and $e$ coincides with the in-neighborhood of some vertex $v$ in the digraph $D$. Any hypergraph with sufficiently many isolated vertices is the competition hypergraph of an acyclic digraph. The hypercompetition number $hk(\\cH)$ of a hypergraph $\\cH$ is defined to be the smallest number of such isolated vertices.\n  In this paper, we study the hypercompetition numbers of hypergraphs. First, we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.5622","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:19:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CaknEZeI3KBgfu+L2B5InubEQV+iPTUcQtKTUbd5B5WYROjnEpVCZKxgMV7dpvD0iRmAKFQojjPwk/g9q/ZfCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T04:57:53.989140Z"},"content_sha256":"0d4dc978e083241838de9f05da23962ff38ed41f275cd8e8793e93e2b2c68924","schema_version":"1.0","event_id":"sha256:0d4dc978e083241838de9f05da23962ff38ed41f275cd8e8793e93e2b2c68924"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JE5PGAI4EHROZC5SEISZ5COKSZ/bundle.json","state_url":"https://pith.science/pith/JE5PGAI4EHROZC5SEISZ5COKSZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JE5PGAI4EHROZC5SEISZ5COKSZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-04T04:57:53Z","links":{"resolver":"https://pith.science/pith/JE5PGAI4EHROZC5SEISZ5COKSZ","bundle":"https://pith.science/pith/JE5PGAI4EHROZC5SEISZ5COKSZ/bundle.json","state":"https://pith.science/pith/JE5PGAI4EHROZC5SEISZ5COKSZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JE5PGAI4EHROZC5SEISZ5COKSZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:JE5PGAI4EHROZC5SEISZ5COKSZ","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":"1363801cf083cd0ca8d899a120226cdd24242f551781db3076c48645aa81727e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-31T08:51:53Z","title_canon_sha256":"202c2fa38392265af49c76a02a307e49189bb9e50ca732ba6c038a467eafcd1b"},"schema_version":"1.0","source":{"id":"1005.5622","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.5622","created_at":"2026-05-18T04:19:33Z"},{"alias_kind":"arxiv_version","alias_value":"1005.5622v3","created_at":"2026-05-18T04:19:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.5622","created_at":"2026-05-18T04:19:33Z"},{"alias_kind":"pith_short_12","alias_value":"JE5PGAI4EHRO","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"JE5PGAI4EHROZC5S","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"JE5PGAI4","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:0d4dc978e083241838de9f05da23962ff38ed41f275cd8e8793e93e2b2c68924","target":"graph","created_at":"2026-05-18T04:19:33Z","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 competition hypergraph $C{\\cH}(D)$ of a digraph $D$ is the hypergraph such that the vertex set is the same as $D$ and $e \\subseteq V(D)$ is a hyperedge if and only if $e$ contains at least 2 vertices and $e$ coincides with the in-neighborhood of some vertex $v$ in the digraph $D$. Any hypergraph with sufficiently many isolated vertices is the competition hypergraph of an acyclic digraph. The hypercompetition number $hk(\\cH)$ of a hypergraph $\\cH$ is defined to be the smallest number of such isolated vertices.\n  In this paper, we study the hypercompetition numbers of hypergraphs. First, we ","authors_text":"Boram Park, Yoshio Sano","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-31T08:51:53Z","title":"On the hypercompetition numbers of hypergraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.5622","kind":"arxiv","version":3},"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":"sha256:2630642192cb0bb2b92a5edb0f9ca153e8c40240c932a344d8585a126cb433da","target":"record","created_at":"2026-05-18T04:19:33Z","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":"1363801cf083cd0ca8d899a120226cdd24242f551781db3076c48645aa81727e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-31T08:51:53Z","title_canon_sha256":"202c2fa38392265af49c76a02a307e49189bb9e50ca732ba6c038a467eafcd1b"},"schema_version":"1.0","source":{"id":"1005.5622","kind":"arxiv","version":3}},"canonical_sha256":"493af3011c21e2ec8bb222259e89ca965360f79046edac834688027113a4fceb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"493af3011c21e2ec8bb222259e89ca965360f79046edac834688027113a4fceb","first_computed_at":"2026-05-18T04:19:33.510659Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:19:33.510659Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cGlm8gkm+/AnzMmpElbvF3JMWbsAAl1FbVMFoE9CCWLGLS0cozLV2xaCGgXitj1OvadP87knaS5gTq7JZS9xDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:19:33.511056Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.5622","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2630642192cb0bb2b92a5edb0f9ca153e8c40240c932a344d8585a126cb433da","sha256:0d4dc978e083241838de9f05da23962ff38ed41f275cd8e8793e93e2b2c68924"],"state_sha256":"dd21341c5149347c20853e113431a9a5d91e81e65d304db35e6a8791f7a0282e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jy694PhGXTrmUK/UADjGpCHImxOIna40lP2/nIwJo13BF2cHjL4X38Ci0TAiEHx7UakZ+Z5Km88UK+Y8+bkbCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T04:57:53.991940Z","bundle_sha256":"7b1883a14bdd8129226a0617c901a39b948437d65589b89894c532877be87df3"}}