Abstract
The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable in P, then |h L (x)–h L (y)|≤k, where h L (x) is the height of x in L. Tanenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem of characterizing the posets of linear discrepancy 2. We show that this problem is equivalent to finding the posets with linear discrepancy equal to 3 having the property that the deletion of any point results in a reduction in the linear discrepancy. Howard determined that there are infinitely many such posets of width 2. We complete the forbidden subposet characterization of posets with linear discrepancy equal to 2 by finding the minimal posets of width 3 with linear discrepancy equal to 3. We do so by showing that, with a small number of exceptions, they can all be derived from the list for width 2 by the removal of specific comparisons.
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The first and second authors were supported during this research by National Science Foundation VIGRE grant DMS-0135290.
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Howard, D.M., Keller, M.T. & Young, S.J. A Characterization of Partially Ordered Sets with Linear Discrepancy Equal to 2. Order 24, 139–153 (2007). https://doi.org/10.1007/s11083-007-9065-1
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DOI: https://doi.org/10.1007/s11083-007-9065-1