{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:BYJDAGOJNWNRG3TQQRGZC62NJC","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":"c88562c14c68b5a26107072a573b2b247f2e3521f6a4a945994b48fbdb454223","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-21T12:48:01Z","title_canon_sha256":"f559c43efd0b295839aa8b20838b918c5d05fd44898850e9a817b846c1230e19"},"schema_version":"1.0","source":{"id":"1812.09089","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.09089","created_at":"2026-05-17T23:57:30Z"},{"alias_kind":"arxiv_version","alias_value":"1812.09089v2","created_at":"2026-05-17T23:57:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.09089","created_at":"2026-05-17T23:57:30Z"},{"alias_kind":"pith_short_12","alias_value":"BYJDAGOJNWNR","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"BYJDAGOJNWNRG3TQ","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"BYJDAGOJ","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:27c904dac42d30368981fcc5322b3397300a0782e3a247b805c4ea9347e9e7af","target":"graph","created_at":"2026-05-17T23:57:30Z","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 a graph on $n$ vertices. A linear forest is a graph consisting of vertex-disjoint paths and isolated vertices. A maximum linear forest of $G$ is a subgraph of $G$ with maximum number of edges, which is a linear forest. We denote by $l(G)$ this maximum number. Let $t=\\left\\lfloor (k-1)/2\\right \\rfloor$. Recently, Ning and Wang \\cite{boning} proved that if $l(G)=k-1$, then for any $k<n$ \\[ e(G) \\leq \\max \\left\\{\\binom{k}{2},\\binom{t}{2}+t (n - t)+ c \\right\\}, \\] where $c=0$ if $k$ is odd and $c=1$ otherwise, and the inequality is tight. In this paper, we prove that if $l(G)=k-1$ and $","authors_text":"Jian Wang, Weihua Yang, Xiuzhuan Duan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-21T12:48:01Z","title":"The maximum number of triangles in graphs without large linear forests"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.09089","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":"sha256:3bb024426b7b29d84f3d2c0d38766eba48bea5ca0a6b87679b6617523da1d783","target":"record","created_at":"2026-05-17T23:57:30Z","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":"c88562c14c68b5a26107072a573b2b247f2e3521f6a4a945994b48fbdb454223","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-21T12:48:01Z","title_canon_sha256":"f559c43efd0b295839aa8b20838b918c5d05fd44898850e9a817b846c1230e19"},"schema_version":"1.0","source":{"id":"1812.09089","kind":"arxiv","version":2}},"canonical_sha256":"0e123019c96d9b136e70844d917b4d4899bf2afd4a9792e34be180e719acc1b4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0e123019c96d9b136e70844d917b4d4899bf2afd4a9792e34be180e719acc1b4","first_computed_at":"2026-05-17T23:57:30.396594Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:30.396594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4Oh+/l/4eLKA1CahURScw7zsy0YiFs5C4HuAzyXY1+TYVQGqPMJ4BerCX/Fe2k1DvkD18udK4S6UgkEb9EW1Bw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:30.397117Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.09089","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3bb024426b7b29d84f3d2c0d38766eba48bea5ca0a6b87679b6617523da1d783","sha256:27c904dac42d30368981fcc5322b3397300a0782e3a247b805c4ea9347e9e7af"],"state_sha256":"5699cf5567ad5bcd4226516e73a10e7e68ee1d75e3e4634bd41b2528a4a8ab71"}