{"paper":{"title":"On the Number of Discrete Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Eyvindur Ari Palsson, Steven Senger","submitted_at":"2019-02-21T20:54:28Z","abstract_excerpt":"We study a generalization of Erd\\H os's unit distances problem to chains of $k$ distances. Given $\\mathcal P,$ a set of $n$ points, and a sequence of distances $(\\delta_1,\\ldots,\\delta_k)$, we study the maximum possible number of tuples of distinct points $(p_1,\\ldots,p_{k+1})\\in \\mathcal P^{k+1}$ satisfying $|p_j p_{j+1}|=\\delta_j$ for every $1\\leq j \\leq k$. We study the problem in $\\mathbb R^2$ and in $\\mathbb R^3$, and derive upper and lower bounds for this family of problems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.08259","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"}