Graphs and Combinatorics

, Volume 34, Issue 2, pp 355–364 | Cite as

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

Original Paper
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Abstract

Let \({\mathbb {Z}}\) be the set of integers and \({\mathbb {Z}}_l\) be the set of integers modulo l. A set \(L\subseteq T={\mathbb {Z}}_{l_1}\times {\mathbb {Z}}_{l_2}\times \cdots \times Z_{l_m}\) is called a line if there exist \({\mathbf {a}},{\mathbf {b}}\in T\) such that \(L=\{ {\mathbf {a}}+t{\mathbf {b}}\in T\ :\ t\in {\mathbb {Z}} \}\). A set \(X\subseteq T\) is called a no-three-in-line set if \(\vert X\cap L\vert \le 2\) for all the lines L in T. The maximum size of a no-three-in-line set is denoted by \(\tau \left( T \right) \). Let \(m\ge 2\) and \(k_1,k_2,\ldots ,k_m\) be positive integers such that \(\gcd (k_i,k_j)=1\) for all ij with \(i\ne j\). In this paper, we will show that
$$\begin{aligned} \tau \left( {\mathbb {Z}}_{k_1n}\times {\mathbb {Z}}_{k_2n}\times \cdots \times Z_{k_mn} \right) \le 2n^{m-1}. \end{aligned}$$
We will give sufficient conditions for which the equality holds. When \(k_1=k_2=\cdots =k_m=1\) and \(n=p^l\) where p is a prime and \(l\ge 1\) is an integer, we will show that equality holds if and only if \(p=2\) and \(l=1\), i.e., \(\tau \left( {\mathbb {Z}}_{p^l}\times {\mathbb {Z}}_{p^l}\times \cdots \times Z_{p^l} \right) =2p^{l(m-1)}\) if and only if \(p=2\) and \(l=1\).

Keywords

Discrete torus No-three-in-line problem Chinese remainder theorem 

Mathematics Subject Classification

05B30 11D79 

Notes

Acknowledgements

We would like to thank the anonymous referee for the comments that had helped us make several improvements to this paper.

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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  2. 2.Institute of Mathematical SciencesUniversity of MalayaKuala LumpurMalaysia

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