Tiling transitive tournaments and their blow-ups

Abstract

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 $h \geq 1$. The following result is proved: Let $c_2=1/2$, $c_3=5/6$ and $c_k=1-2^{-k-\log k}$ for $k \geq 4$. For every $\epsilon > 0$ there exists $N=N(\epsilon,h,k)$ such that for every undirected graph $G$ with $n > N$ vertices and with $\delta(G) \geq c_kn$, every orientation of $G$ contains vertex disjoint copies of $TT(h,k)$ that cover all but at most $\epsilon n$ vertices. In the cases $k=2$ and $k=3$ the result is asymptotically tight. For $k \geq 4$, $c_k$ cannot be improved to less than $1-2^{-0.5k(1+o(1))}$.