Domination in transitive colorings of tournaments

An edge coloring of a tournament T with colors 1, 2, . . . , k is called ktransitive if the digraph T (i) defined by the edges of color i is transitively oriented for each 1 ≤ i ≤ k. We explore a conjecture of the first author: For each positive integer k there exists a (least) p(k) such that every k-transitive tournament has a dominating set of at most 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 [15] (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(22 d−1 d log d) on the d-dimensional boxcover 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.