Inducibility in H-free graphs and inducibility of Turán graphs

For graphs F and H, let i(F) denote the inducibility of F and let i_H(F) denote the inducibility of F over H-free graphs. We prove that for almost all graphs F on a given number of vertices, i_Kk(F) attains infinitely many values as k varies. For complete partite graphs F (and, more generally, for symmetrizable families of graphs F), we prove that i_H(F)=i_Kk(F) where k=χ(H), and is attained by a complete ℓ-partite graphon W_F,k, where ℓ<k. We determine the part sizes of W_F,k for all k, whence determine i(F), whenever F is the Turán graph on s vertices and r parts, for all s≤3r+1, which was recently proved by Liu, Mubayi, and Reiher for s=r+1. As a corollary, this determines the inducibility of all Turán graphs on at most 14 vertices. Furthermore, since inducibility is invariant under complement, this determines the inducibility of all matchings and, more generally, all graphs with maximum degree 1, of any size. Similarly, this determines the inducibility of all triangle factors, of any size. For complete partite graphs F with at most one singleton part, we prove that i_Kk(F) only attains finitely many values as k varies; in particular, there exists t=t(F) such that i(F) is attained by some complete t-partite graphon. This is best possible as it was shown by Liu, Pikhurko, Sharifzadeh, and Staden that this is not necessarily true if there are two singleton parts. Finally, for every r, we give a nontrivial sufficient condition for a complete r-partite graph F to have the property that i(F) is attained by a complete partite graphon all whose part sizes are distinct.