M ar 2 00 6 Mediated Digraphs and Quantum Nonlocality ∗

A digraph D = (V,A) is mediated if for each pair x, y of distinct vertices of D, either xy ∈ A or yx ∈ A or there is a vertex z such that both xz, yz ∈ A. For a digraph D, ∆−(D) is the maximum in-degree of a vertex in D. The nth mediation number μ(n) is the minimum of ∆−(D) over all mediated digraphs on n vertices. Mediated digraphs and μ(n) are of interest in the study of quantum nonlocality. We obtain a lower bound f(n) for μ(n) and determine infinite sequences of values of n for which μ(n) = f(n) and μ(n) > f(n), respectively. We derive upper bounds for μ(n) and prove that μ(n) = f(n)(1+ o(1)). We conjecture that there is a constant c such that μ(n) ≤ f(n)+ c. Methods and results of design theory and number theory are used.