The linear discrepancy of ${3}\times{3}\times{3}$
Commun. Korean Math. Soc. 2010 Vol. 25, No. 1, 19-25
https://doi.org/10.4134/CKMS.2010.25.1.19
Printed March 1, 2010
Gab-Byoung Chae, Minseok Cheong, and Sang-Mok Kim
Wonkwang University, Sogang University, and Kwangwoon University
Abstract : $\mathbf{3} \times \mathbf{3} \times \mathbf{3}$ is the meaningful smallest product of three chains of each size $2n+1$ since $\mathbf{1} \times \mathbf{1} \times \mathbf{1}$ is a 1-element poset. The linear discrepancy of the product of three chains $\mathbf{2n}\times \mathbf{2n}\times \mathbf{2n}$ is found as $6n^3-2n^2-1$. But the case of the product of three chains $\mathbf{(2n+1)}\times \mathbf{(2n+1)}\times \mathbf{(2n+1)}$ is not known yet. In this paper, we determine $ld(\mathbf{3} \times \mathbf{3} \times \mathbf{3})$ as a case to determine the linear discrepancy of the product of three chains of each size $2n+1$.
Keywords : poset, linear discrepancy
MSC numbers : 06A07
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