Fractional Weak Discrepancy of Posets and Certain Forbidden Configurations
نویسندگان
چکیده
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 wdF (P ) of a poset P = (V,≺) was introduced in [7] 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. Semiorders were characterized by their fractional weak discrepancy in [8]. Here we describe the range of values of wdF (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. ∗Supported in part by a Wellesley College Brachman Hoffman Fellowship.
منابع مشابه
Forbidden subposets for fractional weak discrepancy at most k
The fractional weak discrepancy of a poset P , written wdF (P ), is the least k such that some f : P → R satisfies f(y)− f(x) ≥ 1 for x ≺ y and |f(y)− f(x)| ≤ k for x‖y. We determine the minimal forbidden subposets for the property wdF (P ) ≤ k when k is an integer.
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