{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KNULKXAXTIZXEIBOJPDPWE3KI2","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":"ba59b7b0e28ba46ca87f82029fdbb845f0a3cc6c6f8c6f5067cceae13d850adc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-23T07:48:51Z","title_canon_sha256":"1c33531e7767982ff339b3dc6bd024f9b70da5e4cf302a5182aef1f541980d3f"},"schema_version":"1.0","source":{"id":"1307.5967","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.5967","created_at":"2026-05-18T00:28:27Z"},{"alias_kind":"arxiv_version","alias_value":"1307.5967v2","created_at":"2026-05-18T00:28:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.5967","created_at":"2026-05-18T00:28:27Z"},{"alias_kind":"pith_short_12","alias_value":"KNULKXAXTIZX","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"KNULKXAXTIZXEIBO","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"KNULKXAX","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:cf52b9d66fab30f9c97fabba0546a4649c5faf8f390c750ad46ea09ce10678eb","target":"graph","created_at":"2026-05-18T00:28:27Z","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":"Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erd\\H{o}s and R\\'enyi, and the family of $H$-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph $H$. A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical $H$-free graph with $n$ vertices and $m$ edges changes as $m$ grows from $0$ to $\\text{ex}(n,H)$. In this paper, we resolve this problem in the case when $H$ is a clique, extending a classical re","authors_text":"J\\'ozsef Balogh, Lutz Warnke, Robert Morris, Wojciech Samotij","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-23T07:48:51Z","title":"The typical structure of sparse $K_{r+1}$-free graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5967","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":"sha256:4d9ab2b52d0650f0d8058541aa55adddd3f9bf458822071f3321dfd553b19e3c","target":"record","created_at":"2026-05-18T00:28:27Z","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":"ba59b7b0e28ba46ca87f82029fdbb845f0a3cc6c6f8c6f5067cceae13d850adc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-23T07:48:51Z","title_canon_sha256":"1c33531e7767982ff339b3dc6bd024f9b70da5e4cf302a5182aef1f541980d3f"},"schema_version":"1.0","source":{"id":"1307.5967","kind":"arxiv","version":2}},"canonical_sha256":"5368b55c179a3372202e4bc6fb136a4693cde03b0885182ce6b2aa61a3600981","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5368b55c179a3372202e4bc6fb136a4693cde03b0885182ce6b2aa61a3600981","first_computed_at":"2026-05-18T00:28:27.719777Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:28:27.719777Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bY1hTN2dW8eWVuadCeTNZcYNK5a3S331ynHW5ZLmZiOMC4SpgkKhClMyrtmAjwTwHhLmy12cZZ+FgE1r1HQVBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:28:27.720509Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.5967","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4d9ab2b52d0650f0d8058541aa55adddd3f9bf458822071f3321dfd553b19e3c","sha256:cf52b9d66fab30f9c97fabba0546a4649c5faf8f390c750ad46ea09ce10678eb"],"state_sha256":"50fdecc13a8d94f6d04a3cd3348ebe0f80341c06385a61ca52f18f84df129bf6"}