The linear discrepancy of a product of two posets
Bull. Korean Math. Soc. 2017 Vol. 54, No. 3, 1081-1094
https://doi.org/10.4134/BKMS.b160501
Published online May 31, 2017
Minseok Cheong
Korea University
Abstract : For a poset $P=(X, \le_P)$, the linear discrepancy of $P$ is the minimum value of maximal differences of all incomparable elements for all possible labelings. In this paper, we find a lower bound and an upper bound of the linear discrepancy of a product of two posets. In order to give a lower bound, we use the known result, $\ld(\mathbf{m} \times \mathbf{n}) = \left\lceil \frac{mn}{2} \right\rceil -2$. Next, we use Dilworth's chain decomposition to obtain an upper bound of the linear discrepancy of a product of a poset and a chain. Finally, we give an example touching this upper bound.
Keywords : poset, product of posets, linear discrepancy
MSC numbers : 06A07
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