The feasible region of induced graphs

Understand

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.