报告题目：Feasible Region for Hypergraphs

报告人：刘西之 （Department of Mathematics, Statistic and Computer Science，University of Illinois at Chicago）

时间：6月18日（周二）下午 3:00-4:00

地点：1208

摘要：Fix $r\ge 3$. Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $\mathcal{H}$ is $\mathcal{F}$-free if it does not contain any $r$-graph in $\mathcal{F}$ as a subgraph. We define the \textbf{feasible region} $\Omega(\mathcal{F})$ of $\mathcal{F}$ as \[\Omega(\mathcal{F}) = \left\{\left(\lim_{n\to \infty}\frac{|\partial \mathcal{H}|}{\binom{n}{r-1}}, \lim_{n\to \infty}\frac{|\mathcal{H}|}{\binom{n}{r-1}} \right): \text{$\mathcal{H}$ is an $n$-vertex $\mathcal{F}$-free $r$-graph} \right\}\].

The classical Tur\'{a}n problem studies the value $\lim_{n\to \infty}\frac{ex(n,\mathcal{F})}{\binom{n}{r-1}}$, which in our language is equivalent to the maximum value of the projection of $\Omega(F)$ on the $y$-axis. On the other hand, if we let $\mathcal{F}= \emptyset$, then the study of the boundary of $\Omega(\emptyset)$ is equivalent to the celebrated Kruskal-Katona Shadow Theorem.

In this talk, I will first present some general results about $\Omega(\mathcal{F})$.Then, several examples of $\Omega(\mathcal{F})$ will be given. In the end, I will introduce you some related open problems. This is joint work with Dhruv Mubayi.