{ "id": "1902.08259", "version": "v1", "published": "2019-02-21T20:54:28.000Z", "updated": "2019-02-21T20:54:28.000Z", "title": "On the Number of Discrete Chains", "authors": [ "Eyvindur Ari Palsson", "Steven Senger", "Adam Sheffer" ], "comment": "9 pages, 1 figure", "categories": [ "math.CO" ], "abstract": "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.", "revisions": [ { "version": "v1", "updated": "2019-02-21T20:54:28.000Z" } ], "analyses": { "subjects": [ "52C10", "51A20" ], "keywords": [ "discrete chains", "oss unit distances problem", "distinct points", "lower bounds", "derive upper" ], "note": { "typesetting": "TeX", "pages": 9, "language": "en", "license": "arXiv", "status": "editable" } } }