On No-Three-In-Line Problem on m-Dimensional Torus

• Main Authors: Ku, Cheng Yeaw, Wong, Kok Bin
• Format: Article
• Published: Springer Verlag 2018
• Subjects: Q Science (General); QA Mathematics
• Online Access: http://eprints.um.edu.my/21579/; https://doi.org/10.1007/s00373-018-1878-8
• Institution: Universiti Malaya

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.