Amalgamation of team tournaments (Q1110531)

Consider a collection of teams \(T_ 1,...,T_ n\) of players all of different strengths. In a tournament between these teams each player of each team plays once against each player of every other team. Such a tournament generates an oriented graph \(G_ n\) in which the arc \(^{\to}\) is present if and only if the players of team \(T_ i\) win a strict majority of their games against the players of team \(T_ j\). It follows from the author's results that for any three oriented graphs \(G_ n\), \(H_ n\), and \(L_ n\) there exist two collections of teams \(T_ 1,...,T_ n\) and \(S_ 1,...,S_ n\) such that the first collection generates \(G_ n\), the second collection generates \(H_ n\), and the amalgamated collection \(T_ 1\cup S_ 1,...,T_ n\cup S_ n\) generates \(L_ n\).