{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:ULAXILEY3TAD5T75FPCIGTL26I","short_pith_number":"pith:ULAXILEY","canonical_record":{"source":{"id":"2403.12016","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-03-18T17:52:07Z","cross_cats_sorted":[],"title_canon_sha256":"eb2118b604c6447fa3837e2311776f0338816787d28abe424b2607752d336a3d","abstract_canon_sha256":"3c0bc26abc0892ae40ba1e670dd52875d08865072b8c65e0791728abc8a86151"},"schema_version":"1.0"},"canonical_sha256":"a2c1742c98dcc03ecffd2bc4834d7af23cbbf1e656b4257e230cc1e71d7e8942","source":{"kind":"arxiv","id":"2403.12016","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.12016","created_at":"2026-07-05T07:57:35Z"},{"alias_kind":"arxiv_version","alias_value":"2403.12016v1","created_at":"2026-07-05T07:57:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.12016","created_at":"2026-07-05T07:57:35Z"},{"alias_kind":"pith_short_12","alias_value":"ULAXILEY3TAD","created_at":"2026-07-05T07:57:35Z"},{"alias_kind":"pith_short_16","alias_value":"ULAXILEY3TAD5T75","created_at":"2026-07-05T07:57:35Z"},{"alias_kind":"pith_short_8","alias_value":"ULAXILEY","created_at":"2026-07-05T07:57:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:ULAXILEY3TAD5T75FPCIGTL26I","target":"record","payload":{"canonical_record":{"source":{"id":"2403.12016","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-03-18T17:52:07Z","cross_cats_sorted":[],"title_canon_sha256":"eb2118b604c6447fa3837e2311776f0338816787d28abe424b2607752d336a3d","abstract_canon_sha256":"3c0bc26abc0892ae40ba1e670dd52875d08865072b8c65e0791728abc8a86151"},"schema_version":"1.0"},"canonical_sha256":"a2c1742c98dcc03ecffd2bc4834d7af23cbbf1e656b4257e230cc1e71d7e8942","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:57:35.197842Z","signature_b64":"IyEX3qfJl1N3DDeJn4R5JKSROYPSJ/jA6MPRb0/XJpX0FG6xmKl3bIMfhKveH91kfNre9jUvCFpQUcZHnqjSDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2c1742c98dcc03ecffd2bc4834d7af23cbbf1e656b4257e230cc1e71d7e8942","last_reissued_at":"2026-07-05T07:57:35.197347Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:57:35.197347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2403.12016","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:57:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Lt+pTD5wUExz1PrsLh+Hg7TxXT/ASPGmgWYbh+OG1qhouwXJnXRQ4VmAcif89jqQIsMA2XQ+hIbzccitCeGBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T23:29:10.007758Z"},"content_sha256":"be3dfd6e4183f37bcb1984b8d665f405970ce8bb37cde19fa90c344b1805a2de","schema_version":"1.0","event_id":"sha256:be3dfd6e4183f37bcb1984b8d665f405970ce8bb37cde19fa90c344b1805a2de"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:ULAXILEY3TAD5T75FPCIGTL26I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ordered and colored subgraph density problems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dhruv Mubayi, Emily Cairncross","submitted_at":"2024-03-18T17:52:07Z","abstract_excerpt":"We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \\in [0, 1]$ is asymptotically maximized by a clique and isolated vertices or its complement. Next, among ordered $n$-vertex graphs with $m$ edges, we determine the maximum and minimum number of copies of a $k$-edge star whose nonleaf vertex is minimum among all vertices of the star. Finally, for $s \\ge 2$, we define a particular $3$-edge-colored complete graph $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.12016","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2403.12016/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T07:57:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KCyhd7f67H/rvTwfulPCU6W4fMiH8RtPhCGZK9wtblHZcPi0izptu+lRpwVg10HGMqu6wcQC/zBejdOrX+cJDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T23:29:10.008078Z"},"content_sha256":"1aa441f0960ca5171777cfe910f218f56011cb42a28f1b363b43ebd788c5c57f","schema_version":"1.0","event_id":"sha256:1aa441f0960ca5171777cfe910f218f56011cb42a28f1b363b43ebd788c5c57f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ULAXILEY3TAD5T75FPCIGTL26I/bundle.json","state_url":"https://pith.science/pith/ULAXILEY3TAD5T75FPCIGTL26I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ULAXILEY3TAD5T75FPCIGTL26I/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":"2026-08-14T23:29:10Z","links":{"resolver":"https://pith.science/pith/ULAXILEY3TAD5T75FPCIGTL26I","bundle":"https://pith.science/pith/ULAXILEY3TAD5T75FPCIGTL26I/bundle.json","state":"https://pith.science/pith/ULAXILEY3TAD5T75FPCIGTL26I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ULAXILEY3TAD5T75FPCIGTL26I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:ULAXILEY3TAD5T75FPCIGTL26I","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":"3c0bc26abc0892ae40ba1e670dd52875d08865072b8c65e0791728abc8a86151","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-03-18T17:52:07Z","title_canon_sha256":"eb2118b604c6447fa3837e2311776f0338816787d28abe424b2607752d336a3d"},"schema_version":"1.0","source":{"id":"2403.12016","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.12016","created_at":"2026-07-05T07:57:35Z"},{"alias_kind":"arxiv_version","alias_value":"2403.12016v1","created_at":"2026-07-05T07:57:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.12016","created_at":"2026-07-05T07:57:35Z"},{"alias_kind":"pith_short_12","alias_value":"ULAXILEY3TAD","created_at":"2026-07-05T07:57:35Z"},{"alias_kind":"pith_short_16","alias_value":"ULAXILEY3TAD5T75","created_at":"2026-07-05T07:57:35Z"},{"alias_kind":"pith_short_8","alias_value":"ULAXILEY","created_at":"2026-07-05T07:57:35Z"}],"graph_snapshots":[{"event_id":"sha256:1aa441f0960ca5171777cfe910f218f56011cb42a28f1b363b43ebd788c5c57f","target":"graph","created_at":"2026-07-05T07:57:35Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2403.12016/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \\in [0, 1]$ is asymptotically maximized by a clique and isolated vertices or its complement. Next, among ordered $n$-vertex graphs with $m$ edges, we determine the maximum and minimum number of copies of a $k$-edge star whose nonleaf vertex is minimum among all vertices of the star. Finally, for $s \\ge 2$, we define a particular $3$-edge-colored complete graph $","authors_text":"Dhruv Mubayi, Emily Cairncross","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-03-18T17:52:07Z","title":"Ordered and colored subgraph density problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.12016","kind":"arxiv","version":1},"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:be3dfd6e4183f37bcb1984b8d665f405970ce8bb37cde19fa90c344b1805a2de","target":"record","created_at":"2026-07-05T07:57:35Z","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":"3c0bc26abc0892ae40ba1e670dd52875d08865072b8c65e0791728abc8a86151","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-03-18T17:52:07Z","title_canon_sha256":"eb2118b604c6447fa3837e2311776f0338816787d28abe424b2607752d336a3d"},"schema_version":"1.0","source":{"id":"2403.12016","kind":"arxiv","version":1}},"canonical_sha256":"a2c1742c98dcc03ecffd2bc4834d7af23cbbf1e656b4257e230cc1e71d7e8942","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a2c1742c98dcc03ecffd2bc4834d7af23cbbf1e656b4257e230cc1e71d7e8942","first_computed_at":"2026-07-05T07:57:35.197347Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:57:35.197347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IyEX3qfJl1N3DDeJn4R5JKSROYPSJ/jA6MPRb0/XJpX0FG6xmKl3bIMfhKveH91kfNre9jUvCFpQUcZHnqjSDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T07:57:35.197842Z","signed_message":"canonical_sha256_bytes"},"source_id":"2403.12016","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be3dfd6e4183f37bcb1984b8d665f405970ce8bb37cde19fa90c344b1805a2de","sha256:1aa441f0960ca5171777cfe910f218f56011cb42a28f1b363b43ebd788c5c57f"],"state_sha256":"7c6017dae15b7caaf5d47240627895d15bcd0e20ada11fd02b316cf01632964e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pRV+y3/2al3LW3A9XkKWURu9nD0JB2yChaxr6UPZxX7Il7/wH4XO6xhfCj6gtMIQC1TlTt8KYEVHxkt9YWxUDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T23:29:10.012146Z","bundle_sha256":"459e05c348a5d5afcb9c7eb9cac4d3a17447fab4495fc58e3f1c10fa547282de"}}