On the Maximum Induced Density of Directed Stars and Related Problems

Abstract

Let $k \ge 3$ be an integer. We prove that the maximum induced density of the $k$-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 $k=3, 4, 5$. 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.