Domination in transitive colorings of tournaments

An edge coloring of a tournament $T$ with colors $1,2,\dots,k$ is called \it $k$-transitive \rm if the digraph $T(i)$ defined by the edges of color $i$ is transitively oriented for each $1\le i \le k$. We explore a conjecture of the second 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 Erd\H os, Sands, Sauer and Woodrow (so the conjecture is interesting even if false). We show that the conjecture implies a stronger conjecture, a possible extension of a result of B\'ar\'any and Lehel on covering point sets by boxes. The principle used leads also to an upper bound $O(2^{2^{d-1}}d\log d)$ 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 $3^{14}$ to 64 and propose possible further improvements through finding the maximum domination number over parity tournaments.