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.
