In this paper we introduce the notion of the fractional weak discrepancy of a poset, building on previous work on weak discrepancy in [?, ?, ?]. The fractional weak discrepancy wd F (P ) of a poset P = (V, ≺) is the minimum nonnegative k for which there exists a function f :We formulate the fractional weak discrepancy problem as a linear program and show how its solution can also be used to calculate the (integral) weak discrepancy. We interpret the dual linear program as a circulation problem in a related directed graph and use this to give a structural characterization of the fractional weak discrepancy of a poset.

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In this paper we describe the range of values that can be taken by the fractional weak discrepancy of a poset subject to forbidden r + s configurations, where r +s = 4. Generalizing previous work on weak discrepancy in [5,12,13], the notion of fractional weak discrepancy wd F (P ) of a poset P = (V, ≺) was introduced in [7] as the minimum nonnegative k for which there exists a function f :Semiorders were characterized by their fractional weak discrepancy in [8]. Here we describe the range of values of wd F (P ) according to whether P contains an induced 2 + 2 and/or an induced 3 + 1. In particular, we prove that the range for an interval order that is not a semiorder (contains a 3 + 1 but no 2 + 2) is the set of rational numbers greater than or equal to one.

A poset P = (X, ≺) is a unit OC interval order if there exists a representation that assigns an open or closed real interval I(x) of unit length to each x ∈ P so that x ≺ y in P precisely when each point of I(x) is less than each point in I(y). In this paper we give a forbidden poset characterization of the class of unit OC interval orders and an efficient algorithm for recognizing the class. The algorithm takes a poset P as input and either produces a representation or returns a forbidden poset induced in P . arXiv:1501.06430v1 [math.CO]