{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:50Z","timestamp":1753893830578,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $S$ be a set of $n$ points in Euclidean $3$-space. Assign to each $x\\in S$ a distance $r(x)&gt;0$, and let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erd\u0151s and Pach (1988) introduced the extremal quantity $f_3(n)=\\max\\sum_{x\\in S}e_r(x,S)$, where the maximum is taken over all $n$-point subsets $S$ of $3$-space and all assignments $r\\colon S\\to(0,\\infty)$ of distances. We show that if the pair $(S,r)$ maximises $f_3(n)$ and $n$ is sufficiently large, then, except for at most $2$ points, $S$ is contained in a circle $\\mathcal{C}$ and the axis of symmetry $\\mathcal{L}$ of $\\mathcal{C}$, and $r(x)$ equals the distance from $x$ to $C$ for each $x\\in S\\cap\\mathcal{L}$. This, together with a new construction, implies that $f_3(n)=n^2\/4 + 5n\/2 + O(1)$.<\/jats:p>","DOI":"10.37236\/8887","type":"journal-article","created":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T04:52:36Z","timestamp":1589518356000},"source":"Crossref","is-referenced-by-count":1,"title":["Favourite Distances in $3$-Space"],"prefix":"10.37236","volume":"27","author":[{"given":"Konrad J.","family":"Swanepoel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,5,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p17\/8074","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p17\/8074","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T04:52:37Z","timestamp":1589518357000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i2p17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,15]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,4,3]]}},"URL":"https:\/\/doi.org\/10.37236\/8887","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,5,15]]},"article-number":"P2.17"}}