{"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. On the other hand, we prove a Ramsey-type result: for ev"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.4731","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}