{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:24:42Z","timestamp":1759335882873,"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>A cube tiling of $\\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\\{[0,1)^d+t:t\\in T\\}$ such that $\\bigcup_{t\\in T}([0,1)^d+t=\\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\\in [d]=\\{1,\\ldots, d\\}$ and $t_i=s_i$ for every $i\\in [d]\\setminus \\{j\\}$. In $1930$, Keller conjectured that in every cube tiling of $\\mathbb{R}^d$ there is a twin pair. Keller's conjecture is true for dimensions $d\\leq 6$ and false for all dimensions $d\\geq 8$. For $d=7$ the conjecture is still open. Let $x\\in \\mathbb{R}^d$, $i\\in [d]$, and let $L(T,x,i)$ be the set of all $i$th coordinates $t_i$ of vectors $t\\in T$ such that $([0,1)^d+t)\\cap ([0,1]^d+x)\\neq \\emptyset$ and $t_i\\leq x_i$. Let $r^-(T)=\\min_{x\\in \\mathbb{R}^d}\\; \\max_{1\\leq i\\leq d}|L(T,x,i)|$ and $r^+(T)=\\max_{x\\in \\mathbb{R}^d}\\; \\max_{1\\leq i\\leq d}|L(T,x,i)|$.\u00a0It is known that if $r^-(T)\\leq 2$ or $r^+(T)\\geq 6$, then Keller's conjecture is true for $d=7$. In the present paper we show that it is also true for $d=7$ if $r^+(T)=5$. Thus, if $[0,1)^d+T$ is a counterexample to Keller's conjecture in dimension seven, then $r^-(T),r^+(T)\\in \\{3,4\\}$.<\/jats:p>","DOI":"10.37236\/4153","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:33:11Z","timestamp":1578684791000},"source":"Crossref","is-referenced-by-count":5,"title":["On Keller\u2019s Conjecture in Dimension Seven"],"prefix":"10.37236","volume":"22","author":[{"given":"Andrzej P.","family":"Kisielewicz","sequence":"first","affiliation":[]},{"given":"Magdalena","family":"\u0141ysakowska","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,1,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p16\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:38:36Z","timestamp":1579239516000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i1p16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1,20]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/4153","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,1,20]]},"article-number":"P1.16"}}