{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:QVBV7YUHAXEC4UJNWWJMTHLC4R","short_pith_number":"pith:QVBV7YUH","canonical_record":{"source":{"id":"1701.00470","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-01-02T18:01:06Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"c63917f196e2c0dd1208718e3f40465e385669523982ceb85aebe2a6715ce893","abstract_canon_sha256":"13f0695e33d73518ae38f821ab9aad02b246fd787377a5d2c2e82239e16e8c64"},"schema_version":"1.0"},"canonical_sha256":"85435fe28705c82e512db592c99d62e453a12701a9c8c956d5ed82faea85555a","source":{"kind":"arxiv","id":"1701.00470","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.00470","created_at":"2026-05-18T00:53:35Z"},{"alias_kind":"arxiv_version","alias_value":"1701.00470v1","created_at":"2026-05-18T00:53:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.00470","created_at":"2026-05-18T00:53:35Z"},{"alias_kind":"pith_short_12","alias_value":"QVBV7YUHAXEC","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"QVBV7YUHAXEC4UJN","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"QVBV7YUH","created_at":"2026-05-18T12:31:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:QVBV7YUHAXEC4UJNWWJMTHLC4R","target":"record","payload":{"canonical_record":{"source":{"id":"1701.00470","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-01-02T18:01:06Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"c63917f196e2c0dd1208718e3f40465e385669523982ceb85aebe2a6715ce893","abstract_canon_sha256":"13f0695e33d73518ae38f821ab9aad02b246fd787377a5d2c2e82239e16e8c64"},"schema_version":"1.0"},"canonical_sha256":"85435fe28705c82e512db592c99d62e453a12701a9c8c956d5ed82faea85555a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:35.518729Z","signature_b64":"h+MPy4p7lwR3w7uDHBbQ0tkIMTtnLD5Rr4s1L14ipT2eMNZOJA+kBTyxiTJzEbi1HypqdNuzUoPScRGTCPgwBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85435fe28705c82e512db592c99d62e453a12701a9c8c956d5ed82faea85555a","last_reissued_at":"2026-05-18T00:53:35.518225Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:35.518225Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1701.00470","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":"2026-05-18T00:53:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7o74QfhPVYQLVQdj7llZsDHafS3VT2hmDxM1l3vytWPzrpFhXePvKqMRaUfNcTb9YHrMKmJf0zJz8mzyp707Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:07:36.467374Z"},"content_sha256":"a6b16df4f166fda61275fe660fad7820cb05a3115858ff7247620010f200b5fb","schema_version":"1.0","event_id":"sha256:a6b16df4f166fda61275fe660fad7820cb05a3115858ff7247620010f200b5fb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:QVBV7YUHAXEC4UJNWWJMTHLC4R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\text{VC}_{\\ell}$-dimension and the jump to the fastest speed of a hereditary $\\mathcal{L}$-property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.LO","authors_text":"C. Terry","submitted_at":"2017-01-02T18:01:06Z","abstract_excerpt":"In this paper we investigate a connection between the growth rates of certain classes of finite structures and a generalization of $\\text{VC}$-dimension called $\\text{VC}_{\\ell}$-dimension. Let $\\mathcal{L}$ be a finite relational language with maximum arity $r$. A hereditary $\\mathcal{L}$-property is a class of finite $\\mathcal{L}$-structures closed under isomorphism and substructures. The \\emph{speed} of a hereditary $\\mathcal{L}$-property $\\mathcal{H}$ is the function which sends $n$ to $|\\mathcal{H}_n|$, where $\\mathcal{H}_n$ is the set of elements of $\\mathcal{H}$ with universe $\\{1,\\ldot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00470","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":""},"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":"2026-05-18T00:53:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JLO5Na/3wKmm3ECYClOhOGqODEiqei4qiXftrLx1GHuWDBKajC7hkhF9KGIA+8iZeu6RAu7VyejleoTFHa7HDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:07:36.467743Z"},"content_sha256":"c76e8f5685856166cbe2839c23ad7e1b206f00871afdeb91393e637a824b3c39","schema_version":"1.0","event_id":"sha256:c76e8f5685856166cbe2839c23ad7e1b206f00871afdeb91393e637a824b3c39"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QVBV7YUHAXEC4UJNWWJMTHLC4R/bundle.json","state_url":"https://pith.science/pith/QVBV7YUHAXEC4UJNWWJMTHLC4R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QVBV7YUHAXEC4UJNWWJMTHLC4R/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":"2026-06-02T13:07:36Z","links":{"resolver":"https://pith.science/pith/QVBV7YUHAXEC4UJNWWJMTHLC4R","bundle":"https://pith.science/pith/QVBV7YUHAXEC4UJNWWJMTHLC4R/bundle.json","state":"https://pith.science/pith/QVBV7YUHAXEC4UJNWWJMTHLC4R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QVBV7YUHAXEC4UJNWWJMTHLC4R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:QVBV7YUHAXEC4UJNWWJMTHLC4R","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":"13f0695e33d73518ae38f821ab9aad02b246fd787377a5d2c2e82239e16e8c64","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-01-02T18:01:06Z","title_canon_sha256":"c63917f196e2c0dd1208718e3f40465e385669523982ceb85aebe2a6715ce893"},"schema_version":"1.0","source":{"id":"1701.00470","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.00470","created_at":"2026-05-18T00:53:35Z"},{"alias_kind":"arxiv_version","alias_value":"1701.00470v1","created_at":"2026-05-18T00:53:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.00470","created_at":"2026-05-18T00:53:35Z"},{"alias_kind":"pith_short_12","alias_value":"QVBV7YUHAXEC","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"QVBV7YUHAXEC4UJN","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"QVBV7YUH","created_at":"2026-05-18T12:31:39Z"}],"graph_snapshots":[{"event_id":"sha256:c76e8f5685856166cbe2839c23ad7e1b206f00871afdeb91393e637a824b3c39","target":"graph","created_at":"2026-05-18T00:53:35Z","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"},"paper":{"abstract_excerpt":"In this paper we investigate a connection between the growth rates of certain classes of finite structures and a generalization of $\\text{VC}$-dimension called $\\text{VC}_{\\ell}$-dimension. Let $\\mathcal{L}$ be a finite relational language with maximum arity $r$. A hereditary $\\mathcal{L}$-property is a class of finite $\\mathcal{L}$-structures closed under isomorphism and substructures. The \\emph{speed} of a hereditary $\\mathcal{L}$-property $\\mathcal{H}$ is the function which sends $n$ to $|\\mathcal{H}_n|$, where $\\mathcal{H}_n$ is the set of elements of $\\mathcal{H}$ with universe $\\{1,\\ldot","authors_text":"C. Terry","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-01-02T18:01:06Z","title":"$\\text{VC}_{\\ell}$-dimension and the jump to the fastest speed of a hereditary $\\mathcal{L}$-property"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00470","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:a6b16df4f166fda61275fe660fad7820cb05a3115858ff7247620010f200b5fb","target":"record","created_at":"2026-05-18T00:53:35Z","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":"13f0695e33d73518ae38f821ab9aad02b246fd787377a5d2c2e82239e16e8c64","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-01-02T18:01:06Z","title_canon_sha256":"c63917f196e2c0dd1208718e3f40465e385669523982ceb85aebe2a6715ce893"},"schema_version":"1.0","source":{"id":"1701.00470","kind":"arxiv","version":1}},"canonical_sha256":"85435fe28705c82e512db592c99d62e453a12701a9c8c956d5ed82faea85555a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"85435fe28705c82e512db592c99d62e453a12701a9c8c956d5ed82faea85555a","first_computed_at":"2026-05-18T00:53:35.518225Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:35.518225Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h+MPy4p7lwR3w7uDHBbQ0tkIMTtnLD5Rr4s1L14ipT2eMNZOJA+kBTyxiTJzEbi1HypqdNuzUoPScRGTCPgwBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:35.518729Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.00470","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a6b16df4f166fda61275fe660fad7820cb05a3115858ff7247620010f200b5fb","sha256:c76e8f5685856166cbe2839c23ad7e1b206f00871afdeb91393e637a824b3c39"],"state_sha256":"da172d941253b5662ff5593ad390abdd7cf8cda5055f6295cf59c366f9dbc8c0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F+KV9HWKQ4p9VW0694vtu4PjlSDsIwcmWb/f39CHXHzXpsJuE6sBBV72YOtTfeA1q7HYwWd23BHIC2Kh35riAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T13:07:36.469674Z","bundle_sha256":"6bca5548f49a68c97e887150f65cb6b434e0a25c26fd4d0a493ae6226467331e"}}