Largest digraphs contained in all n-tournaments

A directed graph G is an unavoidable subgraph o f all n-tournaments or, simply n-unavoidable, if every tournament on n vertices contain san isomorphic copy o f G, i.e., for each ntournament T there exists an edge preserving injection o f the vertices o f G into the vertices o f T. The problem o f showing certain types o f graphs to be n-unavoidable has been the subject o f several papers, for example, it is known that every ntournament contains a Hamil tonian path ([7]), an antidirected Hamil tonian path ([4]) and a transitive subtournament on [log2 n] vertices ([6]). Results o f this type are also found in [1], [3], and [8]. In this paper we answer the following question: what is the maximum number o f edges that an n-unavoidable subgraph can have? Our graph theoretic terminology is standard. For a vertex v o f a d igraph G=(V, E) we let G+(v)={w[(v, w)EE}. All logari thms are base 2. Let f (n ) (resp. g(n)) be the largest m such that there exists a digraph resp. spanmng, weakly connected diagraph) with m edges that is n-unavoidable subgraph. Trivially f ( n ) ~ g ( n ) . Our main resullt is