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

On No-Three-In-Line Problem on m-Dimensional Torus Let $${\mathbb {Z}}$$ Z be the set of integers and $${\mathbb {Z}}_l$$ Z l be the set of integers modulo l. A set $$L\subseteq T={\mathbb {Z}}_{l_1}\times {\mathbb {Z}}_{l_2}\times \cdots \times Z_{l_m}$$ L ⊆ T = Z l 1 × Z l 2 × ⋯ × Z l m is called a line if there exist $${\mathbf {a}},{\mathbf {b}}\in T$$ a , b ∈ T such that $$L=\{ {\mathbf {a}}+t{\mathbf {b}}\in T\ :\ t\in {\mathbb {Z}} \}$$ L = { a + t b ∈ T : t ∈ Z } . A set $$X\subseteq T$$ X ⊆ T is called a no-three-in-line set if $$\vert X\cap L\vert \le 2$$ | X ∩ L | ≤ 2 for all the lines L in T. The maximum size of a no-three-in-line set is denoted by $$\tau \left( T \right) $$ τ T . Let $$m\ge 2$$ m ≥ 2 and $$k_1,k_2,\ldots ,k_m$$ k 1 , k 2 , … , k m be positive integers such that $$\gcd (k_i,k_j)=1$$ gcd ( k i , k j ) = 1 for all i, j with $$i\ne j$$ i ≠ j . In this paper, we will show that $$\begin{aligned} \tau \left( {\mathbb {Z}}_{k_1n}\times {\mathbb {Z}}_{k_2n}\times \cdots \times Z_{k_mn} \right) \le 2n^{m-1}. \end{aligned}$$ τ Z k 1 n × Z k 2 n × ⋯ × Z k m n ≤ 2 n m - 1 . We will give sufficient conditions for which the equality holds. When $$k_1=k_2=\cdots =k_m=1$$ k 1 = k 2 = ⋯ = k m = 1 and $$n=p^l$$ n = p l where p is a prime and $$l\ge 1$$ l ≥ 1 is an integer, we will show that equality holds if and only if $$p=2$$ p = 2 and $$l=1$$ l = 1 , i.e., $$\tau \left( {\mathbb {Z}}_{p^l}\times {\mathbb {Z}}_{p^l}\times \cdots \times Z_{p^l} \right) =2p^{l(m-1)}$$ τ Z p l × Z p l × ⋯ × Z p l = 2 p l ( m - 1 ) if and only if $$p=2$$ p = 2 and $$l=1$$ l = 1 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Graphs and Combinatorics Springer Journals

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

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Publisher
Springer Japan
Copyright
Copyright © 2018 by Springer Japan KK, part of Springer Nature
Subject
Mathematics; Combinatorics; Engineering Design
ISSN
0911-0119
eISSN
1435-5914
D.O.I.
10.1007/s00373-018-1878-8
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Graphs and CombinatoricsSpringer Journals

Published: Feb 22, 2018

References

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