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 shadow density approaches x and whose edge density approaches y. The feasible region provides a lot of combinatorial information, for example, the supremum of y over all $(x,y)\in \mathrm{\Omega }(\mathcal{F})$ is the Turán density $\pi (\mathcal{F})$, and $\mathrm{\Omega }(\varnothing )$ 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 Bollobás by almost completely determining the feasible region for cancellative triple systems.

让 $\mathcal{F}$是r一致超图的族。可行区域$\mathrm{\Omega }（\mathcal{F}）$ 的 $\mathcal{F}$ 是点集 $（X，ÿ）$ 在单位正方形中，使得存在一系列 $\mathcal{F}$-free ř -uniform超图，其阴影密度接近X和其边缘密度接近ÿ。可行区域提供了很多组合信息，例如，y的全部$（X，ÿ）\in \mathrm{\Omega }（\mathcal{F}）$ 是图兰密度 $\pi （\mathcal{F}）$和 $\mathrm{\Omega }（\varnothing ）$ 给出了Kruskal-Katona定理。

我们对 $\mathrm{\Omega }（\mathcal{F}）$，并证明 $\mathrm{\Omega }（\mathcal{F}）$几乎完全由左连续的微分函数决定；而且，存在一个$\mathcal{F}$为此功能不连续。我们还扩展了一些旧的相关定理。例如，我们将Fisher和Ryan的结果推广到超图，并通过几乎完全确定可取消三元组系统的可行区域来扩展Bollobás的经典结果。