Abstract
A cube tiling of ℝd is a family of pairwise disjoint cubes [0, 1)d + T = {[0, 1)d + t: t ∈ T} such that ∪t∈T([0, 1)d + t) = ℝd. Two cubes [0, 1)d + t, [0, 1)d + s are called a twin pair if |tj−sj| = 1 for some j ∈ [d] = {1, ⋅, d} and ti = si for every i ∈ [d]∖{j}. In 1930, Keller conjectured that in every cube tiling of ℝd there is a twin pair. For x ∈ ℝd and i ∈ [d], let L(T, x, i) be the set of all ith coordinates ti of vectors t ∈ T such that ([0, 1)d + t)∩([0, 1]d + x)≠∅ and ti ≤ xi. Let
Communicated by: M. Joswig
References
[1] J. Brakensiek, M. Heule, J. Mackey, D. Narváez, The resolution of Keller’s conjecture. In: International Joint Conference on Automated Reasoning. Part I, volume 12166 of Lecture Notes in Comput. Sci., 48–65, Springer 2020. MR413980410.1007/978-3-030-51074-9_4Search in Google Scholar
[2] K. Corrádi, S. Szabó, A combinatorial approach for Keller’s conjecture. Period. Math. Hungar. 21 (1990), 95–100. MR1070948 Zbl 0718.5201710.1007/BF01946848Search in Google Scholar
[3] K. Corrádi, S. Szabó, Cube tiling and covering a complete graph. Discrete Math. 85 (1990), 319–321. MR1081840 Zbl 0736.0502410.1016/0012-365X(90)90388-XSearch in Google Scholar
[4] J. Debroni, J. D. Eblen, M. A. Langston, W. Myrvold, P. Shor, D. Weerapurage, A complete resolution of the Keller maximum clique problem. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 129–135, SIAM, Philadelphia, PA 2011. MR2857115 Zbl 1376.0510910.1137/1.9781611973082.11Search in Google Scholar
[5] M. Dutour Sikirić, Y. Itoh, A. Poyarkov, Cube packings, second moment and holes. European J. Combin. 28 (2007), 715–725. MR2300752 Zbl 1111.5201710.1016/j.ejc.2006.01.008Search in Google Scholar
[6] J. A. Grytczuk, A. P. Kisielewicz, K. Przesławski, Minimal partitions of a box into boxes. Combinatorica 24 (2004), 605–614. MR2096817 Zbl 1080.0500810.1007/s00493-004-0037-4Search in Google Scholar
[7] O.-H. Keller, Über die lückenlose Erfüllung des Raumes mit Würfeln. J. Reine Angew. Math. 163 (1930), 231–248. MR1581241 JFM 56.1120.01Search in Google Scholar
[8] A. P. Kisielewicz, Rigid polyboxes and Keller’s conjecture. Adv. Geom. 17 (2017), 203–230. MR3652241 Zbl 1388.5201610.1515/advgeom-2017-0004Search in Google Scholar
[9] A. P. Kisielewicz, Gluing and cutting cube tiling codes in dimension six. Accepted in Discrete Comput. Geom. Preprint 2020, arXiv:2008:10016 [math.CO]Search in Google Scholar
[10] A. P. Kisielewicz, M. Łysakowska, On Keller’s conjecture in dimension seven. Electron. J. Combin. 22 (2015), Paper 1.16, 44 pages. MR3315458 Zbl 1305.0504210.37236/4153Search in Google Scholar
[11] A. P. Kisielewicz, K. Przesławski, Polyboxes, cube tilings and rigidity. Discrete Comput. Geom. 40 (2008), 1–30. MR2429647 Zbl 1147.5200910.1007/s00454-007-9005-2Search in Google Scholar
[12] M. Łysakowska, K. Przesławski, Keller’s conjecture on the existence of columns in cube tilings of ℝn. Adv. Geom. 12 (2012), 329–352. MR2911153 Zbl 1248.5200210.1515/advgeom.2011.055Search in Google Scholar
[13] J. C. Lagarias, P. W. Shor, Keller’s cube-tiling conjecture is false in high dimensions. Bull. Amer. Math. Soc. (N.S.) 27 (1992), 279–283. MR1155280 Zbl 0759.5201310.1090/S0273-0979-1992-00318-XSearch in Google Scholar
[14] J. C. Lagarias, P. W. Shor, Cube-tilings of ℝn and nonlinear codes. Discrete Comput. Geom. 11 (1994), 359–391. MR1273224 Zbl 0804.5201310.1007/BF02574014Search in Google Scholar
[15] J. Lawrence, Tiling ℝd by translates of the orthants. In: Convexity and related combinatorial geometry (Norman, Okla., 1980), 203–207, Dekker 1982. MR650314 Zbl 0483.52012Search in Google Scholar
[16] J. Mackey, A cube tiling of dimension eight with no facesharing. Discrete Comput. Geom. 28 (2002), 275–279. MR1920144 Zbl 1018.5201910.1007/s00454-002-2801-9Search in Google Scholar
[17] O. Perron, Über lückenlose Ausfüllung des n-dimensionalen Raumes durch kongruente Würfel. Math. Z. 46 (1940), 1–26.10.1007/BF01181421Search in Google Scholar
[18] S. Szabó, A reduction of Keller’s conjecture. Period. Math. Hungar. 17 (1986), 265–277. MR866636 Zbl 0613.5200610.1007/BF01848388Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston