{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:THN2TVQIFW65O624IHJJBFV23U","short_pith_number":"pith:THN2TVQI","canonical_record":{"source":{"id":"1512.06371","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-20T13:25:46Z","cross_cats_sorted":[],"title_canon_sha256":"36ffdc28c0e83a1b721584b105a9f60c236e224508e258ccd649f49799b4649c","abstract_canon_sha256":"544839ed495f080076b02878082c51c23c1a62c6825b6cd69d9a6b0319571d5f"},"schema_version":"1.0"},"canonical_sha256":"99dba9d6082dbdd77b5c41d29096badd1d9a7833a65bf2acb92ed1f08693edce","source":{"kind":"arxiv","id":"1512.06371","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.06371","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"arxiv_version","alias_value":"1512.06371v1","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.06371","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"pith_short_12","alias_value":"THN2TVQIFW65","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"THN2TVQIFW65O624","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"THN2TVQI","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:THN2TVQIFW65O624IHJJBFV23U","target":"record","payload":{"canonical_record":{"source":{"id":"1512.06371","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-20T13:25:46Z","cross_cats_sorted":[],"title_canon_sha256":"36ffdc28c0e83a1b721584b105a9f60c236e224508e258ccd649f49799b4649c","abstract_canon_sha256":"544839ed495f080076b02878082c51c23c1a62c6825b6cd69d9a6b0319571d5f"},"schema_version":"1.0"},"canonical_sha256":"99dba9d6082dbdd77b5c41d29096badd1d9a7833a65bf2acb92ed1f08693edce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:00.717040Z","signature_b64":"rqHJ4aIRozp0Axfmr6gKIyQGvK+xVJiwxFDpmT1SikBwSMBXXvP4kBS70usbSSxVxsQUIJJR2KTtN36V70wuDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99dba9d6082dbdd77b5c41d29096badd1d9a7833a65bf2acb92ed1f08693edce","last_reissued_at":"2026-05-18T01:24:00.716389Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:00.716389Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.06371","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":"2026-05-18T01:24:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m8mhaukxb9LvDPgnyKIUIXd2ltnPPc9eCXg2PETLZjbNM1QbCtE8eIkc2odcm+zMQAafDwqqHjACqv7wTIClCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:25:29.369787Z"},"content_sha256":"f19f2f23d7a19966720fc7e5bb5038bf60877e5dfd27fd16975a669962123255","schema_version":"1.0","event_id":"sha256:f19f2f23d7a19966720fc7e5bb5038bf60877e5dfd27fd16975a669962123255"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:THN2TVQIFW65O624IHJJBFV23U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A path Turan problem for infinite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Craig Timmons, Xing Peng","submitted_at":"2015-12-20T13:25:46Z","abstract_excerpt":"Let $G$ be an infinite graph whose vertex set is the set of positive integers, and let $G_n$ be the subgraph of $G$ induced by the vertices $\\{1,2, \\dots , n \\}$. An increasing path of length $k$ in $G$, denoted $I_k$, is a sequence of $k+1$ vertices $1 \\leq i_1 < i_2 < \\dots < i_{k+1}$ such that $i_1, i_2, \\ldots, i_{k+1}$ is a path in $G$. For $k \\geq 2$, let $p(k)$ be the supremum of $\\liminf_{ n \\rightarrow \\infty} \\frac{ e(G_n) }{n^2}$ over all $I_k$-free graphs $G$. In 1962, Czipszer, Erd\\H{o}s, and Hajnal proved that $p(k) = \\frac{1}{4} (1 - \\frac{1}{k})$ for $k \\in \\{2,3 \\}$. Erd\\H{o}s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06371","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":"2026-05-18T01:24:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zmdN5nWwkUFw8aKz1JToURdEB79yp+HYA6FAz3iy4o8D9Wi7iH39wFKAM4vbqxV6ckeTJXRELLhf+Id6cMr9Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:25:29.370131Z"},"content_sha256":"392f7d9e6fe454b6eea2829d6bf48dfe7fa9460998e89358fb997fa2dfa80929","schema_version":"1.0","event_id":"sha256:392f7d9e6fe454b6eea2829d6bf48dfe7fa9460998e89358fb997fa2dfa80929"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/THN2TVQIFW65O624IHJJBFV23U/bundle.json","state_url":"https://pith.science/pith/THN2TVQIFW65O624IHJJBFV23U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/THN2TVQIFW65O624IHJJBFV23U/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":"2026-06-02T22:25:29Z","links":{"resolver":"https://pith.science/pith/THN2TVQIFW65O624IHJJBFV23U","bundle":"https://pith.science/pith/THN2TVQIFW65O624IHJJBFV23U/bundle.json","state":"https://pith.science/pith/THN2TVQIFW65O624IHJJBFV23U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/THN2TVQIFW65O624IHJJBFV23U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:THN2TVQIFW65O624IHJJBFV23U","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":"544839ed495f080076b02878082c51c23c1a62c6825b6cd69d9a6b0319571d5f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-20T13:25:46Z","title_canon_sha256":"36ffdc28c0e83a1b721584b105a9f60c236e224508e258ccd649f49799b4649c"},"schema_version":"1.0","source":{"id":"1512.06371","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.06371","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"arxiv_version","alias_value":"1512.06371v1","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.06371","created_at":"2026-05-18T01:24:00Z"},{"alias_kind":"pith_short_12","alias_value":"THN2TVQIFW65","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"THN2TVQIFW65O624","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"THN2TVQI","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:392f7d9e6fe454b6eea2829d6bf48dfe7fa9460998e89358fb997fa2dfa80929","target":"graph","created_at":"2026-05-18T01:24:00Z","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 $G$ be an infinite graph whose vertex set is the set of positive integers, and let $G_n$ be the subgraph of $G$ induced by the vertices $\\{1,2, \\dots , n \\}$. An increasing path of length $k$ in $G$, denoted $I_k$, is a sequence of $k+1$ vertices $1 \\leq i_1 < i_2 < \\dots < i_{k+1}$ such that $i_1, i_2, \\ldots, i_{k+1}$ is a path in $G$. For $k \\geq 2$, let $p(k)$ be the supremum of $\\liminf_{ n \\rightarrow \\infty} \\frac{ e(G_n) }{n^2}$ over all $I_k$-free graphs $G$. In 1962, Czipszer, Erd\\H{o}s, and Hajnal proved that $p(k) = \\frac{1}{4} (1 - \\frac{1}{k})$ for $k \\in \\{2,3 \\}$. Erd\\H{o}s","authors_text":"Craig Timmons, Xing Peng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-20T13:25:46Z","title":"A path Turan problem for infinite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06371","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:f19f2f23d7a19966720fc7e5bb5038bf60877e5dfd27fd16975a669962123255","target":"record","created_at":"2026-05-18T01:24:00Z","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":"544839ed495f080076b02878082c51c23c1a62c6825b6cd69d9a6b0319571d5f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-20T13:25:46Z","title_canon_sha256":"36ffdc28c0e83a1b721584b105a9f60c236e224508e258ccd649f49799b4649c"},"schema_version":"1.0","source":{"id":"1512.06371","kind":"arxiv","version":1}},"canonical_sha256":"99dba9d6082dbdd77b5c41d29096badd1d9a7833a65bf2acb92ed1f08693edce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99dba9d6082dbdd77b5c41d29096badd1d9a7833a65bf2acb92ed1f08693edce","first_computed_at":"2026-05-18T01:24:00.716389Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:00.716389Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rqHJ4aIRozp0Axfmr6gKIyQGvK+xVJiwxFDpmT1SikBwSMBXXvP4kBS70usbSSxVxsQUIJJR2KTtN36V70wuDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:00.717040Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.06371","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f19f2f23d7a19966720fc7e5bb5038bf60877e5dfd27fd16975a669962123255","sha256:392f7d9e6fe454b6eea2829d6bf48dfe7fa9460998e89358fb997fa2dfa80929"],"state_sha256":"d8b225df1365e9e431e611f00c145c5e5aae5ff3dfe356609824b76acee2da66"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0DCcvCQKFLkL7r7Lnmu8htmjiZNEFrsdKTP9Vt++gCBC+aWO5jVTeuFhob/AKDX77MkxRl0yNEf2xVCtHRuTDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T22:25:29.372102Z","bundle_sha256":"d5c11743bef54023219cb6a4478f28649bb51937b68bb993936692a19b11c3cf"}}