On the Number of 5-Cycles in a Tournament

If T is an n-vertex tournament with a given number of 3-cycles, what can be said about the number of its 4-cycles? The most interesting range of this problem is where T is assumed to have c ⋅ n cyclic triples for some c > 0 and we seek to minimize the number of 4-cycles. We conjecture that the (asymptotic) minimizing T is a random blow-up of a constant-sized transitive tournament. Using the method of flag algebras, we derive a lower bound that almost matches the conjectured value. We are able to answer the easier problem of maximizing the number of 4-cycles. These questions can be equivalently stated in terms of transitive subtournaments. Namely, given the number of transitive triples in T , how many transitive quadruples can it have? As far as we know, this is the first study of inducibility in tournaments.