Tiling Transitive Tournaments and Their Blow-ups

Let TTk denote the transitive tournament on k vertices. Let TT(h,k) denote the graph obtained from TTk 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 ck=1−2−k−log k for k≥4. For every ∈>0 there exists N=N(∈,h,k) such that for every undirected graph G with n>N vertices and with δ(G)≥ckn, every orientation of G contains vertex disjoint copies of TT(h,k) that cover all but at most ∈n vertices. In the cases k=2 and k=3 the result is asymptotically tight. For k≥4, ck cannot be improved to less than 1−2−0.5k(1+o(1)).