Dominant tournament families

For a tournament H with h vertices, its typical density is given by h!2(-(h/2))/aut(H), i.e. this is the expected density of H in a random tournament. A family F of h-vertex tournaments is dominant if for all sufficiently large n, there exists an n-vertex tournament G such that the density of each element of F in G is larger than its typical density by a constant factor. Characterizing all dominant families is challenging already for small h. Here we characterize several large dominant families for every h. In particular, we prove the following for all h sufficiently large: (i) For all tournaments H * with at least 5 log h vertices, the family of all h-vertex tournaments that contain H * as a subgraph is dominant. (ii) The family of all h-vertex tournaments whose minimum feedback arc set size is at most 1/2 (h 2) - h(3/2) root ln h is dominant. For small h, we construct a dominant family of 6 (i.e. 50% of the) tournaments on 5 vertices and dominant families of size larger than 40% for h = 6, 7, 8, 9. For all h, we provide an explicit construction of a dominant family which is conjectured to obtain an absolute constant fraction of the tournaments on h vertices. Some additional intriguing open problems are presented.