THE LINEAR DISCREPANCY OF 3 × 3 × 3

Title & Authors
THE LINEAR DISCREPANCY OF 3 × 3 × 3
Chae, Gab-Byoung; Cheong, Min-Seok; Kim, Sang-Mok;

Abstract
$\small{3{\times}3{\times}3}$ is the meaningful smallest product of three chains of each size 2n+1 since $\small{1{\times}1{\times}1}$ is a 1-element poset. The linear discrepancy of the product of three chains $\small{2n{\times}2n{\times}2n}$ is found as $\small{6n^3-2n^2-1}$. But the case of the product of three chains $\small{(2n + 1){\times}(2n + 1){\times}(2n + 1)}$ is not known yet. In this paper, we determine ld$\small{(3{\times}3{\times}3)}$ as a case to determine the linear discrepancy of the product of three chains of each size 2n + 1.
Keywords
poset;linear discrepancy;
Language
English
Cited by
References
1.
G. -B. Chae, M. Cheong, and S. -M. Kim, Irreducible posets of linear discrepancy 1 and 2, FJMS 22 (2006), no. 2, 217–226.

2.
M. Cheong and S.-M. Kim, The linear discrepancy of the product of three chains of size 2n, FJMS 30 (2008), 285–298.

3.
S. P. Hong, J. Y. Hynn, H. K. Kim, and S.-M. Kim, Linear discrepancy of the product of two chains, Order 22 (2005), 63–72.

4.
P. Fishburn, P. Tanenbaum, and A. Trenk, Linear discrepancy and bandwidth, Order 18 (2001), 237–245.

5.
P. Tanenbaum, A. Trenk, and P. Fishburn, Linear discrepancy and weak discrepancy of partially ordered sets, Order 18 (2001), 201–225.