Journal of Combinatorial Theory Series B ( IF 1.2 ) Pub Date : 2020-12-24 , DOI: 10.1016/j.jctb.2020.12.004 Xizhi Liu , Dhruv Mubayi
Let be a family of r-uniform hypergraphs. The feasible region of is the set of points in the unit square such that there exists a sequence of -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 is the Turán density , and gives the Kruskal-Katona theorem.
We undertake a systematic study of , and prove that is completely determined by a left-continuous almost everywhere differentiable function; and moreover, there exists an 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.
中文翻译:
超图的可行区域
让 是r一致超图的族。可行区域 的 是点集 在单位正方形中,使得存在一系列 -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 是图兰密度 和 给出了Kruskal-Katona定理。
我们对 ,并证明 几乎完全由左连续的微分函数决定;而且,存在一个为此功能不连续。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。