Upper bounds on the size of transitive subtournaments in digraphs

In this paper, we consider upper bounds on the size of transitive subtournaments in a digraph. In particular, we give an upper bound based on linear algebraic techniques similarly to Hoffman's bound for the size of cocliques in a regular graph. Furthermore, we partially improve the bound for doubly regular tournaments by using the technique of Greaves and Soicher for strongly regular graphs [4], which gives a new application of block intersection polynomials.