Tiling transitive tournaments and their blow-ups

Let TT_k denote the transitive tournament on k vertices. Let TT (h, k) denote the graph obtained from TT_k by replacing each vertex with an independent set of size h ≥ 1. The following result is proved: Let c₂ 1/2, c₃ = 5/6 and c_k = 1 - 2 ^-k-logk for k ≥ 4. For every ε > 0 there exists N = N(ε, h, k) such that for every undirected graph G with n > N vertices and with δ(G) ≥ c_kn, every orientation of G contains vertex disjoint copies of TT (h, k) that cover all but at most εn vertices. In the cases k = 2 and k = 3 the result is asymptotically tight. For k ≥ 4, c _k cannot be improved to less than 1 - 2^{-0.5k(1+o(1))}.