The feasible region of induced graphs

The feasible region $\Omega_{{\rm ind}}(F)$ of a graph $F$ is the collection of points $(x,y)$ in the unit square such that there exists a sequence of graphs whose edge densities approach $x$ and whose induced $F$-densities approach $y$. A complete description of $\Omega_{{\rm ind}}(F)$ is not known for any $F$ with at least four vertices that is not a clique or an independent set. The feasible region provides a lot of combinatorial information about $F$. For example, the supremum of $y$ over all $(x,y)\in \Omega_{{\rm ind}}(F)$ is the inducibility of $F$ and $\Omega_{{\rm ind}}(K_r)$ yields the Kruskal-Katona and clique density theorems. We begin a systematic study of $\Omega_{{\rm ind}}(F)$ by proving some general statements about the shape of $\Omega_{{\rm ind}}(F)$ and giving results for some specific graphs $F$. Many of our theorems apply to the more general setting of quantum graphs. For example, we prove a bound for quantum graphs that generalizes an old result of Bollob\'as for the number of cliques in a graph with given edge density. We also consider the problems of determining $\Omega_{{\rm ind}}(F)$ when $F=K_r^-$, $F$ is a star, or $F$ is a complete bipartite graph. In the case of $K_r^-$ our results sharpen those predicted by the edge-statistics conjecture of Alon et. al. while also extending a theorem of Hirst for $K_4^-$ that was proved using computer aided techniques and flag algebras. The case of the 4-cycle seems particularly interesting and we conjecture that $\Omega_{{\rm ind}}(C_4)$ is determined by the solution to the triangle density problem, which has been solved by Razborov.