Degree bounds for linear discrepancy of interval orders and disconnected posets

Abstract

Let P be a poset in which each point is incomparable to at most Δ others. Tanenbaum, Trenk, and Fishburn asked in a 2001 paper if the linear discrepancy of such a poset is bounded above by ⌊ ( 3 Δ − 1 ) / 2 ⌋ . 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 Δ , with equality precisely for interval orders containing an antichain of size Δ + 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 ⌊ ( 3 Δ − 1 ) / 2 ⌋ . 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.