The number of C3⃗-free vertices on 3-partite tournaments

Let TT be a 3-partite tournament. We say that a vertex vv is C3⃗-free if vv does not lie on any directed triangle of TT. Let F3(T)F3(T) be the set of the C3⃗-free vertices in a 3-partite tournament and f3(T)f3(T) its cardinality. In this paper we prove that if TT is a regular 3-partite tournament, then F3(T)F3(T) must be contained in one of the partite sets of TT. It is also shown that for every regular 3-partite tournament, f3(T)f3(T) does not exceed n9, where nn is the order of TT. On the other hand, we give an infinite family of strongly connected tournaments having n−4n−4C3⃗-free vertices. Finally we prove that for every c≥3c≥3 there exists an infinite family of strongly connected cc-partite tournaments, Dc(T)Dc(T), with n−c−1C3⃗-free vertices.