Let $\mathcal{ F}$ be a family of $r$-uniform hypergraphs. The feasible region $\Omega(\mathcal{ F})$ of $\mathcal{ F}$ is the set of points $(x,y)$ in the unit square such that there exists a sequence of $\mathcal{ F}$-free $r$-uniform hypergraphs whose edge density approaches $x$ and whose shadow density approaches $y$. The feasible region provides a lot of combinatorial information, for example, the supremum of $y$ over all $(x,y) \in \Omega(\mathcal{ F})$ is the Tur\'{ a}n density $\pi(\mathcal{ F})$, and $\Omega(\emptyset)$ gives the Kruskal-Katona theorem. We undertake a systematic study of $\Omega(\mathcal{ F})$, and prove that $\Omega(\mathcal{ F})$ is completely determined by a left-continuous almost everywhere differentiable function; and moreover, there exists an $\mathcal{ F}$ for which this function is not continuous. We also extend some old related theorems. For example, we generalize a result of Fisher and Ryan to hypergraphs and extend a classical result of Bollob\'as by almost completely determining the feasible region for cancellative triple systems.
Categories: math.CO