A note on an embedding problem in transitive tournaments

Let TT_n be a transitive tournament on n vertices. It is known Görlich, Pil?niak, Wo?niak, (2006) 3] that for any acyclic oriented graph of order n and size not greater than , two graphs isomorphic to are arc-disjoint subgraphs of TT_n. In this paper, we consider the problem of embedding of acyclic oriented graphs into their complements in transitive tournaments. We show that any acyclic oriented graph of size at most is embeddable into all its complements in TT_n. Moreover, this bound is generally the best possible.