Dominating sets in k-majority tournaments

A k-majority tournament T on a finite vertex set V is defined by a set of 2k − 1 linear orderings of V , with u→ v if and only if u lies above v in at least k of the orders. Motivated in part by the phenomenon of “non-transitive dice”, we let F (k) be the maximum over all k-majority tournaments T of the size of a minimum dominating set of T . We show that F (k) exists for all k > 0, that F (2) = 3 and that in general C1k/ log k ≤ F (k) ≤ C2k log k for suitable positive constants C1 and C2.