Domination in transitive colorings of tournaments

An edge coloring of a tournament T   with colors 1,2,…,k1,2,…,k is called k-transitive   if the digraph T(i)T(i) defined by the edges of color i   is transitively oriented for each 1⩽i⩽k1⩽i⩽k. We explore a conjecture of the second author: For each positive integer k there exists a (least  ) p(k)p(k)such that every k-transitive tournament has a dominating set of at most  p(k)p(k)vertices.We show how this conjecture relates to other conjectures and results. For example, it is a special case of a well-known conjecture of Erdős, Sands, Sauer and Woodrow [14] (so the conjecture is interesting even if false). We show that the conjecture implies a stronger conjecture, a possible extension of a result of Bárány and Lehel on covering point sets by boxes. The principle used leads also to an upper bound O(22d−1dlogd) on the d-dimensional box-cover number that is better than all previous bounds, in a sense close to best possible. We also improve the best bound known in 3 dimensions from 314 to 64 and propose possible further improvements through finding the maximum domination number over parity tournaments.