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These notions have been studied in analysis, approximation theory etc. since the 1940s.\n  We show that 3-monotone interpolability is very non-local: we exhibit an arbitrarily large finite $P$ for which every proper subset is $3$-monotone interpolable but $P$ itself is not. On the other hand, we prove a Ramsey-type result: for ev"},"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":"1404.4731","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2014-04-18T09:39:49Z","cross_cats_sorted":[],"title_canon_sha256":"d9589896b6976c089111674ec056a881a5aae4b5fc32982824434b681b25a0ef","abstract_canon_sha256":"3979afbc2c75c31a51b399702081ee4b6eb567492ffa84d4d6e8bb3209e5af14"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:24.469163Z","signature_b64":"XBNyHV2cF/wobZKAP+AcWNjzrs/bhfIQ8sbjPIP9YLSV/8iMOAmjijLGQK9cMDNDkW7ulI5gl0n7EevLR4FcAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"736669ce54c225695e9e6390d2d4682fb9d1c8e8e8817215cc9e5e10c2f558a4","last_reissued_at":"2026-05-18T01:33:24.468551Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:24.468551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Three-monotone interpolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Ji\\v{r}\\'i Matou\\v{s}ek, Josef Cibulka, Pavel Pat\\'ak","submitted_at":"2014-04-18T09:39:49Z","abstract_excerpt":"A function $f\\colon\\mathbb R\\to\\mathbb R$ is called \\emph{$k$-monotone} if it is $(k-2)$-times differentiable and its $(k-2)$nd derivative is convex. A point set $P\\subset\\mathbb R^2$ is \\emph{$k$-monotone interpolable} if it lies on a graph of a $k$-monotone function. These notions have been studied in analysis, approximation theory etc. since the 1940s.\n  We show that 3-monotone interpolability is very non-local: we exhibit an arbitrarily large finite $P$ for which every proper subset is $3$-monotone interpolable but $P$ itself is not. 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