Fractional weak discrepancy and interval orders

The fractional weak discrepancy wdF (P ) of a poset P = (V,≺) was introduced in [6] as the minimum nonnegative k for which there exists a function f : V → R satisfying (i) if a ≺ b then f(a)+1 ≤ f(b) and (ii) if a ‖ b then |f(a) − f(b)| ≤ k. In this paper we generalize results in [7, 8] on the range of the wdF function for semiorders (interval orders with no induced 3+ 1) to interval orders with no n+ 1, where n ≥ 3. In particular, we prove that the range for such posets P is the set of rationals that can be written as r/s, where 0 ≤ s − 1 ≤ r < (n − 2)s. If wdF (P ) = r/s and P has an optimal forcing cycle C with up(C) = r and side(C) = s, then r ≤ (n − 2)(s − 1). Moreover when s ≥ 2, for each r satisfying s− 1 ≤ r ≤ (n− 2)(s− 1) there is an interval order having such an optimal forcing cycle and containing no n+ 1. ∗Supported in part by a Wellesley College Brachman Hoffman Fellowship.