The number of oriantations having no fixed tournament (Q2495691)

For a fixed tournament \(T\), let \(D(\mathbf G,T)\) be the number of all orientations of an undirected graph \(\mathbf G\) such that they do not contain \(T\) as a subgraph. Let \(D(n,T)\) denote the maximum of \(D(\mathbf G,T)\) through all undirected graphs \(\mathbf G\) on \(n\) vertices. For a natural number \(k\), let \(t_{k-1}(n)\) denote the maximum number of edges of an undirected graph on \(n\) vertices without a clique on \(k\) vertices. It is proved that if a tournament \(T\) is on \(k\) vertices then there exists \(n_0\) such that \(D(n,T)=2^{t_{k-1}(n)}\) for all \(n\geq n_0\). Some consequences of this result are derived and some open problems are formulated.