Degree bounds for linear discrepancy of interval orders and disconnected posets

Let P be a poset in which each point is incomparable to at most ∆ others. Tanenbaum, Trenk, and Fishburn asked in a 2001 paper if the linear discrepancy of such a poset is bounded above by b(3∆−1)/2c. This paper answers their question in the affirmative for two classes of posets. The first class is the interval orders, which are shown to have linear discrepancy at most ∆, with equality precisely for interval orders containing an antichain of size ∆+ 1. The stronger bound is tight even for interval orders of width 2. The second class of posets considered is the disconnected posets, which have linear discrepancy at most b(3∆−1)/2c. This paper also contains lemmas on the role of critical pairs in linear discrepancy as well as a theorem establishing that every poset contains a point whose removal decreases the linear discrepancy by at most 1.