The feasible region of hypergraphs

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. The feasible region $\mathrm{\Omega }(\mathcal{F})$ 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}$)$istheTur\acute{a}ndensity$\pi($\mathcal{F}$)$,and$\Omega(\emptyset)$ gives the Kruskal-Katona theorem. We undertake a systematic study of $\mathrm{\Omega }(\mathcal{F})$, and prove that $\mathrm{\Omega }(\mathcal{F})$ 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.