Rainbow EKR with stability: From hypercube to Hamming graph
数学专题报告
报告题目(Title)：Rainbow EKR with stability: From hypercube to Hamming graph
报告人(Speaker)：吴耀琨 教授 （上海交通大学）
地点(Place)：腾讯会议 ID：176361275
时间(Time)：2022 年 12 月 14 日 (周三), 19:30--20:30
邀请人(Inviter)：吕本建、王恺顺
报告摘要
We have $n$ disjoint sets $S_1,\ldots,S_n$. A rainbow set is a set contained in $\cup_{i=1}^n S_i$ and containing one or zero point from each $S_i$. A $t$-sunflower is a family of rainbow sets for which a common rainbow $t$-set is contained by every member of this family.
A $t$-intersecting family is a family of rainbow sets any two of which have at least $t$ elements in common. Assign a nonnegative weight $\mu (x)$ for each $x\in \cup_{i=1}^n S_i$ so that
$\sum_{x\in S_i} \mu (x)\leq 1$ holds for $i=1,\ldots, n$. For each $i\in [n]$, let $p_i=1-\sum_{x\in S_i}\mu (x)$. For each rainbow set $X$, define its measure to be $\prod_{S_i\cap X=\emptyset}p_i\prod_{x\in X}\mu (x)$. For any family of rainbow sets, its measure is the sum of the measure of its members.
Under certain technical assumptions, we conjecture that the maximum measure of $t$-intersecting rainbow set family is achieved by a $t$-sunflower. We further conjecture that if a $t$-intersecting rainbow set family has measure close to that maximum measure, then its symmetric difference with a maximum measure $t$-sunflower has a measure close to zero. We verify these conjectures for $t=1$.
This is joint work with Anyuan Tian.
主讲人简介
吴耀琨，上海交大博士毕业后留校工作至今。在上海交大任教期间努力借助各种上课机会认真学习一些简单有趣的数学，并且与学生一起将一些教学体会写成研究文章。
