Graphs and Combinatorics

, Volume 34, Issue 2, pp 355–364

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

• Cheng Yeaw Ku
• Kok Bin Wong
Original Paper

## 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

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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