{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:46Z","timestamp":1753893826694,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In 1946, Paul Erd\u0151s posed a problem of determining the largest possible cardinality of an isosceles set, i.e., a set of points in plane or in space, any three of which form an isosceles triangle. Such a question can be asked for any metric space, and an upper bound ${n+2\\choose 2}$ for the Euclidean space ${\\Bbb E}^{n}$ was found by Blokhuis.  This upper bound is known to be sharp for $n=1$, 2, 6, and 8.  We will consider Erd\u0151s' question for the binary Hamming space $H_{n}$ and obtain the following upper bounds on the cardinality of an isosceles subset $S$ of $H_{n}$: if there are at most two distinct nonzero distances between points of $S$, then $|S|\\leq{n+1\\choose 2}+1$; if, furthermore, $n\\geq 4$, $n\\ne 6$, and, as a set of vertices of the $n$-cube, $S$ is contained in a hyperplane, then $|S|\\leq{n\\choose 2}$; if there are more than two distinct nonzero distances between points of $S$, then $|S|\\leq{n\\choose 2}+1$.  The first bound is sharp if and only if $n=2$ or $n=5$; the other two bounds are sharp for all relevant values of $n$, except the third bound for $n=6$, when the sharp upper bound is 12.  We also give the exact answer to the Erd\u0151s problem for ${\\Bbb E}^{n}$ with $n\\leq 7$ and describe all isosceles sets of the largest cardinality in these dimensions.<\/jats:p>","DOI":"10.37236\/230","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:23:46Z","timestamp":1578716626000},"source":"Crossref","is-referenced-by-count":0,"title":["Isosceles Sets"],"prefix":"10.37236","volume":"16","author":[{"given":"Yury J.","family":"Ionin","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,11,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r141\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r141\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:03:58Z","timestamp":1579305838000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r141"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,11,24]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/230","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,11,24]]},"article-number":"R141"}}