{"paper":{"title":"Determining a Points Configuration from a Subset of the Pairwise Distances","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.MG","authors_text":"Elad Tzalik, Itai Benjamini","submitted_at":"2022-08-29T19:53:09Z","abstract_excerpt":"We study rigidity without assuming general position. Given $n$ distinct labelled points and a set $\\mathcal{P}\\subseteq \\binom{[n]}{2}$ of revealed pairs, we ask when the corresponding distances determine the configuration up to isometry. On the line, we prove an extremal result: if $|\\mathcal{P}|=\\Omega(n^{3/2})$, then there is an induced globally rigid subgraph on $\\Omega(|\\mathcal{P}|/n)$ vertices. In other words, any dense enough graph will contain a subset of labels whose locations can be determined from their distances up to isometry. To prove this, we establish a graph-theoretic result,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.13855","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2208.13855/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}