报告题目：On non-empty cross-intersecting families
报 告 人：钱建国，厦门大学 教授
报告摘要：In this talk we introduce some new results on extremal set theory. Let $2^{[n]}$ and $\binom{[n]}{i}$ be the power set and the collection of all $i$-subsets of $\{1,2,\ldots,n\}$, respectively. We call $t$ ($t\geq2$) families $\mathscr{A}_1,\mathscr{A}_2,\ldots, \mathscr{A}_t\subseteq 2^{[n]}$ {\it cross-intersecting} if $A_i\cap A_j\neq \emptyset$ for any $A_i\in \mathscr{A}_i$ and $A_j\in \mathscr{A}_j$ with $i\neq j$. We show that, for $n\geq k+l, l\geq r\geq 1, c>0$ and $\mathscr{A}\subseteq \binom{[n]}{k},\mathscr{B}\subseteq \binom{[n]}{l}$, if $\mathscr{A}$ and $\mathscr{B}$ are cross-intersecting and $\binom{n-r}{l-r}\leq|\mathscr{B}|\leq \binom{n-1}{l-1}$, then
$$|\mathscr{A}|+c|\mathscr{B}|\leq \max\left\{\binom{n}{k}-\binom{n-r}{k}+c\binom{n-r}{l-r},\ \binom{n-1}{k-1}+c\binom{n-1}{l-1}\right\}.$$ This implies a result of Tokushige and Frankl, and also yields that, for $n\geq 2k$, if $\mathscr{A}_1,\mathscr{A}_2,\ldots,\mathscr{A}_t$ $\subseteq \binom{[n]}{k}$ are non-empty cross-intersecting, then $$\sum\limits_{i=1}^t|\mathscr{A}_i|\leq \max\left\{\binom{n}{k}-\binom{n-k}{k}+t-1,\ t\binom{n-1}{k-1}\right\},$$ which generalizes the corresponding result of Hilton and Milner for $t=2$. Moreover, the extremal families attaining the two upper bounds above are also characterized.
报告人简介：钱建国，厦门大学教授，博士生导师，主要从事图论与组合计数方面的研究。1998年在四川大学获得理学博士学位，现任中国运筹学会图论组合分会常务理事，福建省运筹学会副理事长。主持和参加多项国家面上及重点基金项目，在Combinatorica, J. Combin. Theory, Ser A, Ser B及 J. Graph Theory等国内外学术期刊上发表论文70余篇。
报告时间：2022年12月16日10:00-12:00
报告地点：腾讯会议ID 264-280-614
主办单位：伟德国际1946源自英国
