On No-Three-In-Line Problem on m-Dimensional Torus
Let Z be the set of integers and Zl be the set of integers modulo l. A set L⊆T=Zl1×Zl2Zlm is called a line if there exist a, b∈ T such that L={a+tb∈T:t∈Z}. A set X⊆ T is called a no-three-in-line set if | X∩ L| ≤ 2 for all the lines L in T. The maximum size of a no-three-in-line set is denoted by τ(...
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| Format: | Article |
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Springer Verlag
2018
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| _version_ | 1825721711445671936 |
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| author | Ku, Cheng Yeaw Wong, Kok Bin |
| author_facet | Ku, Cheng Yeaw Wong, Kok Bin |
| author_sort | Ku, Cheng Yeaw |
| collection | UM |
| description | Let Z be the set of integers and Zl be the set of integers modulo l. A set L⊆T=Zl1×Zl2Zlm is called a line if there exist a, b∈ T such that L={a+tb∈T:t∈Z}. A set X⊆ T is called a no-three-in-line set if | X∩ L| ≤ 2 for all the lines L in T. The maximum size of a no-three-in-line set is denoted by τ(T). Let m≥ 2 and k1, k2, … , km be positive integers such that gcd (ki, kj) = 1 for all i, j with i≠ j. In this paper, we will show that (Formula presented.).We will give sufficient conditions for which the equality holds. When k1= k2= ⋯ = km= 1 and n= pl where p is a prime and l≥ 1 is an integer, we will show that equality holds if and only if p= 2 and l= 1 , i.e., τ(Zpl×Zpl×⋯×Zpl) = 2pl(m-1) if and only if p= 2 and l= 1. |
| first_indexed | 2024-03-06T05:54:31Z |
| format | Article |
| id | um.eprints-21579 |
| institution | Universiti Malaya |
| last_indexed | 2024-03-06T05:54:31Z |
| publishDate | 2018 |
| publisher | Springer Verlag |
| record_format | dspace |
| spelling | um.eprints-215792019-07-04T08:58:55Z http://eprints.um.edu.my/21579/ On No-Three-In-Line Problem on m-Dimensional Torus Ku, Cheng Yeaw Wong, Kok Bin Q Science (General) QA Mathematics Let Z be the set of integers and Zl be the set of integers modulo l. A set L⊆T=Zl1×Zl2Zlm is called a line if there exist a, b∈ T such that L={a+tb∈T:t∈Z}. A set X⊆ T is called a no-three-in-line set if | X∩ L| ≤ 2 for all the lines L in T. The maximum size of a no-three-in-line set is denoted by τ(T). Let m≥ 2 and k1, k2, … , km be positive integers such that gcd (ki, kj) = 1 for all i, j with i≠ j. In this paper, we will show that (Formula presented.).We will give sufficient conditions for which the equality holds. When k1= k2= ⋯ = km= 1 and n= pl where p is a prime and l≥ 1 is an integer, we will show that equality holds if and only if p= 2 and l= 1 , i.e., τ(Zpl×Zpl×⋯×Zpl) = 2pl(m-1) if and only if p= 2 and l= 1. Springer Verlag 2018 Article PeerReviewed Ku, Cheng Yeaw and Wong, Kok Bin (2018) On No-Three-In-Line Problem on m-Dimensional Torus. Graphs and Combinatorics, 34 (2). pp. 355-364. ISSN 0911-0119, DOI https://doi.org/10.1007/s00373-018-1878-8 <https://doi.org/10.1007/s00373-018-1878-8>. https://doi.org/10.1007/s00373-018-1878-8 doi:10.1007/s00373-018-1878-8 |
| spellingShingle | Q Science (General) QA Mathematics Ku, Cheng Yeaw Wong, Kok Bin On No-Three-In-Line Problem on m-Dimensional Torus |
| title | On No-Three-In-Line Problem on m-Dimensional Torus |
| title_full | On No-Three-In-Line Problem on m-Dimensional Torus |
| title_fullStr | On No-Three-In-Line Problem on m-Dimensional Torus |
| title_full_unstemmed | On No-Three-In-Line Problem on m-Dimensional Torus |
| title_short | On No-Three-In-Line Problem on m-Dimensional Torus |
| title_sort | on no three in line problem on m dimensional torus |
| topic | Q Science (General) QA Mathematics |
| work_keys_str_mv | AT kuchengyeaw onnothreeinlineproblemonmdimensionaltorus AT wongkokbin onnothreeinlineproblemonmdimensionaltorus |