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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 $"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1812.09089","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-21T12:48:01Z","cross_cats_sorted":[],"title_canon_sha256":"f559c43efd0b295839aa8b20838b918c5d05fd44898850e9a817b846c1230e19","abstract_canon_sha256":"c88562c14c68b5a26107072a573b2b247f2e3521f6a4a945994b48fbdb454223"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:30.397117Z","signature_b64":"4Oh+/l/4eLKA1CahURScw7zsy0YiFs5C4HuAzyXY1+TYVQGqPMJ4BerCX/Fe2k1DvkD18udK4S6UgkEb9EW1Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e123019c96d9b136e70844d917b4d4899bf2afd4a9792e34be180e719acc1b4","last_reissued_at":"2026-05-17T23:57:30.396594Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:30.396594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The maximum number of triangles in graphs without large linear forests","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jian Wang, Weihua Yang, Xiuzhuan Duan","submitted_at":"2018-12-21T12:48:01Z","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. 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