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 $2n+1$.