{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:NNIML2P427KCLU4V4XHGAEI7PZ","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":"6cd127bca08d233c9cf76bf25fd0363c9528e513f173a3c4573683eb4cc13be1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-03T05:48:28Z","title_canon_sha256":"692cd93eb658117c42959ba51e7a2bca5d607ef27f5b189bc05460f53229b8d3"},"schema_version":"1.0","source":{"id":"1702.00912","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.00912","created_at":"2026-05-18T00:43:54Z"},{"alias_kind":"arxiv_version","alias_value":"1702.00912v2","created_at":"2026-05-18T00:43:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.00912","created_at":"2026-05-18T00:43:54Z"},{"alias_kind":"pith_short_12","alias_value":"NNIML2P427KC","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_16","alias_value":"NNIML2P427KCLU4V","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_8","alias_value":"NNIML2P4","created_at":"2026-05-18T12:31:34Z"}],"graph_snapshots":[{"event_id":"sha256:4cc1b1956a80709318adaf1ebbdfe0c280ef32afc02930df5002f8ed001c6906","target":"graph","created_at":"2026-05-18T00:43:54Z","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":"Let $X$ be a finite collection of sets (or \"clusters\"). We consider the problem of counting the number of ways a cluster $A \\in X$ can be partitioned into two disjoint clusters $A_1, A_2 \\in X$, thus $A = A_1 \\uplus A_2$ is the disjoint union of $A_1$ and $A_2$; this problem arises in the run time analysis of the ASTRAL algorithm in phylogenetic reconstruction. We obtain the bound $$ | \\{ (A_1,A_2,A) \\in X \\times X \\times X: A = A_1 \\uplus A_2 \\} | \\leq |X|^{3/p} $$ where $|X|$ denotes the cardinality of $X$, and $p := \\log_3 \\frac{27}{4} = 1.73814\\dots$, so that $\\frac{3}{p} = 1.72598\\dots$. ","authors_text":"Daniel Kane, Terence Tao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-03T05:48:28Z","title":"A bound on partitioning clusters"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.00912","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":"sha256:2bcf3688cb1352395066e590c0286ff732dbb2d4fd725d33cb0757cf7b7a484c","target":"record","created_at":"2026-05-18T00:43:54Z","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":"6cd127bca08d233c9cf76bf25fd0363c9528e513f173a3c4573683eb4cc13be1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-03T05:48:28Z","title_canon_sha256":"692cd93eb658117c42959ba51e7a2bca5d607ef27f5b189bc05460f53229b8d3"},"schema_version":"1.0","source":{"id":"1702.00912","kind":"arxiv","version":2}},"canonical_sha256":"6b50c5e9fcd7d425d395e5ce60111f7e62a3e17f239ba148763483ca07a82e8b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6b50c5e9fcd7d425d395e5ce60111f7e62a3e17f239ba148763483ca07a82e8b","first_computed_at":"2026-05-18T00:43:54.961141Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:54.961141Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wBo5AA8fpOi6qee14mkh6PuzKZT9VZ6IR8rhvg+jWVxSfv2y3iKToKSyJbdFwCn8MKqpm2Ie7EWgAIP2JFUYBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:54.961667Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.00912","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2bcf3688cb1352395066e590c0286ff732dbb2d4fd725d33cb0757cf7b7a484c","sha256:4cc1b1956a80709318adaf1ebbdfe0c280ef32afc02930df5002f8ed001c6906"],"state_sha256":"143325bb72865c1b7be2390284f236727d5ee325f504ac880784a75de50c6982"}