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Basic questions are to determine when $ex_{\\rightarrow}(n,F)$ and $ex_{\\cir}(n,F)$ are linear in $n$, the latter posed by Bra\\ss-K\\'arolyi-Valtr in 2003. In this paper, we answer both these questions for every tree $F$.\n  We give a forbidden subgraph characterization for a family $\\cal T$ of ordered trees with $k$ edges, and show that $ex_{\\rightarrow}(n,T) = (k - 1)n - {k \\choose 2}$ for all $n \\geq k + 1$ when $"},"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":"1812.05750","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-14T01:01:01Z","cross_cats_sorted":[],"title_canon_sha256":"0b417664c96992375580cb41b7c9adc9b6b0ee606a77c456f7a2589ad80118e6","abstract_canon_sha256":"c4ad253b6be8ff68744664f2f7be15ce0bf780e4d3b1240b8742118e9a32bf23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:58.584405Z","signature_b64":"NbPOuSQPYfBP2XRilGSZjA14Zi+txwyfyQHp1GxqCd5UTkhRR8PK1tCSCP85Ofc2CXN84pvNdkMdA6uyrX3BAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81b1fa1af92c5ce952a46eb4a83652c8c5ddf8ba89bbc7190102870b740414b7","last_reissued_at":"2026-05-17T23:54:58.583777Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:58.583777Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ordered and convex geometric trees with linear extremal function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Kostochka, Dhruv Mubayi, Jacques Verstra\\\"ete, Zolt\\'an F\\\"uredi","submitted_at":"2018-12-14T01:01:01Z","abstract_excerpt":"The extremal functions $ex_{\\rightarrow}(n,F)$ and $ex_{\\cir}(n,F)$ for ordered and convex geometric acyclic graphs $F$ have been extensively investigated by a number of researchers. 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