{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:FU7YBFX52XBNY43MGKD74WH5TS","short_pith_number":"pith:FU7YBFX5","canonical_record":{"source":{"id":"1708.01822","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-05T22:10:48Z","cross_cats_sorted":[],"title_canon_sha256":"087a2fc9e4b9ecac184e1ae2df57199370eb7a368127227a1a0edd67c10e2d85","abstract_canon_sha256":"8f28496e9da5846f91cdfe3fae73cd85b717bdab1fabb5bd84afdde5843acac3"},"schema_version":"1.0"},"canonical_sha256":"2d3f8096fdd5c2dc736c3287fe58fd9ca23d7df774101274202e19aa8f6b7a75","source":{"kind":"arxiv","id":"1708.01822","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.01822","created_at":"2026-05-18T00:14:06Z"},{"alias_kind":"arxiv_version","alias_value":"1708.01822v3","created_at":"2026-05-18T00:14:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01822","created_at":"2026-05-18T00:14:06Z"},{"alias_kind":"pith_short_12","alias_value":"FU7YBFX52XBN","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FU7YBFX52XBNY43M","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FU7YBFX5","created_at":"2026-05-18T12:31:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:FU7YBFX52XBNY43MGKD74WH5TS","target":"record","payload":{"canonical_record":{"source":{"id":"1708.01822","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-05T22:10:48Z","cross_cats_sorted":[],"title_canon_sha256":"087a2fc9e4b9ecac184e1ae2df57199370eb7a368127227a1a0edd67c10e2d85","abstract_canon_sha256":"8f28496e9da5846f91cdfe3fae73cd85b717bdab1fabb5bd84afdde5843acac3"},"schema_version":"1.0"},"canonical_sha256":"2d3f8096fdd5c2dc736c3287fe58fd9ca23d7df774101274202e19aa8f6b7a75","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:06.716227Z","signature_b64":"/AwOqf2CAOkCOIl+DRaZgeeKJ4/cBAq5eU4k2xukYVD+/oJh8zymX68v/fiBljr4leWH9Q54tdpXCFXJ2KwFAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d3f8096fdd5c2dc736c3287fe58fd9ca23d7df774101274202e19aa8f6b7a75","last_reissued_at":"2026-05-18T00:14:06.715708Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:06.715708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1708.01822","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":"2026-05-18T00:14:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DODOZyOV/cZU1cjm8itvaat11B5m6FbxlN+wrDmp4wCqUIzR7wfaPfHGgmYG4zzF6CUEBmty8zDxRsFAlY/9Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T08:03:37.962368Z"},"content_sha256":"17d39f11ab7ca5ee2cc793aef902c8b7912d6461bcb08ab4cde32a4f6d679330","schema_version":"1.0","event_id":"sha256:17d39f11ab7ca5ee2cc793aef902c8b7912d6461bcb08ab4cde32a4f6d679330"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:FU7YBFX52XBNY43MGKD74WH5TS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximum star densities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Reiher, Stephan Wagner","submitted_at":"2017-08-05T22:10:48Z","abstract_excerpt":"Given an integer $k \\geq 2$ and a real number $\\gamma\\in [0, 1]$, which graphs of edge density $\\gamma$ contain the largest number of $k$-edge stars? For $k=2$ Ahlswede and Katona proved that asymptotically there cannot be more such stars than in a clique or in the complement of a clique (depending on the value of $\\gamma$). Here we extend their result to all integers $k\\ge 2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01822","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":"2026-05-18T00:14:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FgVgRqzvICRWZQw7xhzQvtLnX/MxElxRtGLXL9ucnagYQwoTB0FuGkiS7n/rIqRsDeomFLOJmVAoMTdLyvn5Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T08:03:37.963073Z"},"content_sha256":"7e6e7c63b4d3767bc113c67c6073e891ca8826e6954088ff04416591cdef04b2","schema_version":"1.0","event_id":"sha256:7e6e7c63b4d3767bc113c67c6073e891ca8826e6954088ff04416591cdef04b2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FU7YBFX52XBNY43MGKD74WH5TS/bundle.json","state_url":"https://pith.science/pith/FU7YBFX52XBNY43MGKD74WH5TS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FU7YBFX52XBNY43MGKD74WH5TS/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":"2026-06-09T08:03:37Z","links":{"resolver":"https://pith.science/pith/FU7YBFX52XBNY43MGKD74WH5TS","bundle":"https://pith.science/pith/FU7YBFX52XBNY43MGKD74WH5TS/bundle.json","state":"https://pith.science/pith/FU7YBFX52XBNY43MGKD74WH5TS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FU7YBFX52XBNY43MGKD74WH5TS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:FU7YBFX52XBNY43MGKD74WH5TS","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":"8f28496e9da5846f91cdfe3fae73cd85b717bdab1fabb5bd84afdde5843acac3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-05T22:10:48Z","title_canon_sha256":"087a2fc9e4b9ecac184e1ae2df57199370eb7a368127227a1a0edd67c10e2d85"},"schema_version":"1.0","source":{"id":"1708.01822","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.01822","created_at":"2026-05-18T00:14:06Z"},{"alias_kind":"arxiv_version","alias_value":"1708.01822v3","created_at":"2026-05-18T00:14:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01822","created_at":"2026-05-18T00:14:06Z"},{"alias_kind":"pith_short_12","alias_value":"FU7YBFX52XBN","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FU7YBFX52XBNY43M","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FU7YBFX5","created_at":"2026-05-18T12:31:15Z"}],"graph_snapshots":[{"event_id":"sha256:7e6e7c63b4d3767bc113c67c6073e891ca8826e6954088ff04416591cdef04b2","target":"graph","created_at":"2026-05-18T00:14:06Z","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":"Given an integer $k \\geq 2$ and a real number $\\gamma\\in [0, 1]$, which graphs of edge density $\\gamma$ contain the largest number of $k$-edge stars? For $k=2$ Ahlswede and Katona proved that asymptotically there cannot be more such stars than in a clique or in the complement of a clique (depending on the value of $\\gamma$). Here we extend their result to all integers $k\\ge 2$.","authors_text":"Christian Reiher, Stephan Wagner","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-05T22:10:48Z","title":"Maximum star densities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01822","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":"sha256:17d39f11ab7ca5ee2cc793aef902c8b7912d6461bcb08ab4cde32a4f6d679330","target":"record","created_at":"2026-05-18T00:14:06Z","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":"8f28496e9da5846f91cdfe3fae73cd85b717bdab1fabb5bd84afdde5843acac3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-05T22:10:48Z","title_canon_sha256":"087a2fc9e4b9ecac184e1ae2df57199370eb7a368127227a1a0edd67c10e2d85"},"schema_version":"1.0","source":{"id":"1708.01822","kind":"arxiv","version":3}},"canonical_sha256":"2d3f8096fdd5c2dc736c3287fe58fd9ca23d7df774101274202e19aa8f6b7a75","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2d3f8096fdd5c2dc736c3287fe58fd9ca23d7df774101274202e19aa8f6b7a75","first_computed_at":"2026-05-18T00:14:06.715708Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:06.715708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/AwOqf2CAOkCOIl+DRaZgeeKJ4/cBAq5eU4k2xukYVD+/oJh8zymX68v/fiBljr4leWH9Q54tdpXCFXJ2KwFAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:06.716227Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.01822","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:17d39f11ab7ca5ee2cc793aef902c8b7912d6461bcb08ab4cde32a4f6d679330","sha256:7e6e7c63b4d3767bc113c67c6073e891ca8826e6954088ff04416591cdef04b2"],"state_sha256":"c19602b5b750f5c8c2170a2ba8f8ac1bd5f31063d81638c3f322fccc6c00f010"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FCmy98vSqLZRcoypT3Yi2Hd/yGgS/3sq7HGiTDgLX7zTlC0ZNloCvUlOmWU1B1sElSfmcJUb145ABA63Px+jDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T08:03:37.966903Z","bundle_sha256":"292fa054ac3f78e5fb2692efccdd95105617e881b417aae4f1a07f0701e7fae8"}}