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