On the Number of 4-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 (Formula presented.) cyclic triples for some (Formula presented.) 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.