On the maximum induced density of directed stars and related problems

Let κ ≥ 3 be an integer. We prove that the maximum induced density of the κ-vertex directed star in a directed graph is asymptotically attained by an iterated blow-up construction. This confirms a conjecture of Falgas-Ravry and Vaughan, who proved this for κ = 3, 4, 5. This question provides the first explicitly known instance of a density problem for which one can prove extremality of an iterated blow-up construction. We also study the inducibility of complete bipartite digraphs and discuss other related problems.