Partition problems in high dimensional boxes
Alon, Bohman, Holzman and Kleitman proved that any partition of a d-dimensional discrete box into proper sub-boxes must consist of at least 2d sub-boxes. Recently, Leader, Milićević and Tan considered the question of how many odd-sized proper boxes are needed to partition a d-dimensional box of odd...
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| Published in | Journal of combinatorial theory. Series A Vol. 166; pp. 315 - 336 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
01.08.2019
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0097-3165 1096-0899 1096-0899 |
| DOI | 10.1016/j.jcta.2019.02.011 |
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| Abstract | Alon, Bohman, Holzman and Kleitman proved that any partition of a d-dimensional discrete box into proper sub-boxes must consist of at least 2d sub-boxes. Recently, Leader, Milićević and Tan considered the question of how many odd-sized proper boxes are needed to partition a d-dimensional box of odd size, and they asked whether the trivial construction consisting of 3d boxes is best possible. We show that approximately 2.93d boxes are enough, and consider some natural generalisations. |
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| AbstractList | Alon, Bohman, Holzman and Kleitman proved that any partition of a d-dimensional discrete box into proper sub-boxes must consist of at least 2d sub-boxes. Recently, Leader, Milićević and Tan considered the question of how many odd-sized proper boxes are needed to partition a d-dimensional box of odd size, and they asked whether the trivial construction consisting of 3d boxes is best possible. We show that approximately 2.93d boxes are enough, and consider some natural generalisations. |
| Author | Long, Jason Bucic, Matija Lidický, Bernard Wagner, Adam Zsolt |
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| References | Kleitman (br0080) 1993; 9 Kearnes, Kiss (br0060) 1999; 207 Saks (br0100) 2002; 257.2–3 Ahlswede, Yudin (br0010) 2006; vol. 4123 Kisielewicz, Przesławski (br0070) 2008; 40 Leader, Milićević, Tan (br0090) 2018; 154 Fekete (br0030) 1923; 17 Grytczuk, Kisielewicz, Przesławski (br0040) 2004; 24 (br0050) 2018 Alon, Bohman, Holzman, Kleitman (br0020) 2002; 257 Alon (10.1016/j.jcta.2019.02.011_br0020) 2002; 257 Kisielewicz (10.1016/j.jcta.2019.02.011_br0070) 2008; 40 Ahlswede (10.1016/j.jcta.2019.02.011_br0010) 2006; vol. 4123 Kleitman (10.1016/j.jcta.2019.02.011_br0080) 1993; 9 Grytczuk (10.1016/j.jcta.2019.02.011_br0040) 2004; 24 Kearnes (10.1016/j.jcta.2019.02.011_br0060) 1999; 207 Leader (10.1016/j.jcta.2019.02.011_br0090) 2018; 154 Saks (10.1016/j.jcta.2019.02.011_br0100) 2002; 257.2–3 Fekete (10.1016/j.jcta.2019.02.011_br0030) 1923; 17 |
| References_xml | – volume: 257 start-page: 255 year: 2002 end-page: 258 ident: br0020 article-title: On partitions of discrete boxes publication-title: Discrete Math. – year: 2018 ident: br0050 – volume: 24 start-page: 605 year: 2004 end-page: 614 ident: br0040 article-title: Minimal partitions of a box into boxes publication-title: Combinatorica – volume: 9 start-page: 95 year: 1993 end-page: 107 ident: br0080 article-title: Partitioning a rectangle into many sub-rectangles so that a line can meet only a few publication-title: DIMACS Ser. Discrete Math. Theoret. Comput. Sci. – volume: 207 start-page: 89 year: 1999 end-page: 135 ident: br0060 article-title: Finite algebras of finite complexity publication-title: Discrete Math. – volume: 257.2–3 start-page: 225 year: 2002 end-page: 247 ident: br0100 article-title: Kleitman and combinatorics publication-title: Discrete Math. – volume: 17 start-page: 228 year: 1923 end-page: 249 ident: br0030 article-title: Über der Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten publication-title: Math. Z. – volume: 40 start-page: 1 year: 2008 end-page: 30 ident: br0070 article-title: Polyboxes, cube tilings and rigidity publication-title: Discrete Comput. Geom. – volume: vol. 4123 year: 2006 ident: br0010 article-title: On partitions of a rectangle into rectangles with restricted number of cross sections publication-title: General Theory of Information Transfer and Combinatorics – volume: 154 start-page: 21 year: 2018 end-page: 31 ident: br0090 article-title: Decomposing the complete publication-title: J. Combin. Theory Ser. A – volume: 257 start-page: 255 year: 2002 ident: 10.1016/j.jcta.2019.02.011_br0020 article-title: On partitions of discrete boxes publication-title: Discrete Math. doi: 10.1016/S0012-365X(02)00428-4 – volume: 207 start-page: 89 year: 1999 ident: 10.1016/j.jcta.2019.02.011_br0060 article-title: Finite algebras of finite complexity publication-title: Discrete Math. doi: 10.1016/S0012-365X(99)00042-4 – volume: 24 start-page: 605 year: 2004 ident: 10.1016/j.jcta.2019.02.011_br0040 article-title: Minimal partitions of a box into boxes publication-title: Combinatorica doi: 10.1007/s00493-004-0037-4 – volume: 9 start-page: 95 year: 1993 ident: 10.1016/j.jcta.2019.02.011_br0080 article-title: Partitioning a rectangle into many sub-rectangles so that a line can meet only a few publication-title: DIMACS Ser. Discrete Math. Theoret. Comput. Sci. doi: 10.1090/dimacs/009/09 – volume: 40 start-page: 1 year: 2008 ident: 10.1016/j.jcta.2019.02.011_br0070 article-title: Polyboxes, cube tilings and rigidity publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-007-9005-2 – volume: 257.2–3 start-page: 225 year: 2002 ident: 10.1016/j.jcta.2019.02.011_br0100 article-title: Kleitman and combinatorics publication-title: Discrete Math. doi: 10.1016/S0012-365X(02)00596-4 – volume: 17 start-page: 228 year: 1923 ident: 10.1016/j.jcta.2019.02.011_br0030 article-title: Über der Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten publication-title: Math. Z. doi: 10.1007/BF01504345 – volume: vol. 4123 year: 2006 ident: 10.1016/j.jcta.2019.02.011_br0010 article-title: On partitions of a rectangle into rectangles with restricted number of cross sections – volume: 154 start-page: 21 year: 2018 ident: 10.1016/j.jcta.2019.02.011_br0090 article-title: Decomposing the complete r-graph publication-title: J. Combin. Theory Ser. A doi: 10.1016/j.jcta.2017.08.008 |
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| Title | Partition problems in high dimensional boxes |
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