The feasible region of induced graphs

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