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We determine ex(n, kP_3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex (n, kP_l) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erd\\H{o}s-S\\'os conjecture, and conditional on its truth we determine ex(n;H) when H is an equibipartite forest, for appropriately l"},"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":"1106.5904","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-06-29T10:57:56Z","cross_cats_sorted":[],"title_canon_sha256":"78b20b3602863ca2a3f771cf513e33f165e5a1df5d2e8c394378092def9c6e9c","abstract_canon_sha256":"586ae27fc01ffd495c10441ea88486d51086263445e59ef7b6d7d4e346fbe72a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:26.898100Z","signature_b64":"stnVcz/gAlzts0HRZVurttLwPTZGjsz+xRUKMFxngD+vPh9h60rqTZJMbQhGUIPAiSmFXmK3MoVjeDdxh7LYDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e26dbc18bfc1566499bacc5098a1a31073506141a65946c4ed2efec8db056176","last_reissued_at":"2026-05-18T03:57:26.897376Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:26.897376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tur\\`an numbers of Multiple Paths and Equibipartite Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nathan Kettle, Neal Bushaw","submitted_at":"2011-06-29T10:57:56Z","abstract_excerpt":"The Tur\\'an number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. 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