报告人：Shagnik Das, Freie Universitat Berlin,Germany

报告时间：8月14日 16:30-17:30

报告地点：1208

摘要：Given a family $/mathcal{F}$ of subsets of $[n]$, we say two sets $A, B /in /mathcal{F}$ are comparable if $A /subset B$ or $B /subset A$. Sperner’s celebrated theorem gives the size of the largest family without any comparable pairs. This result was later generalised by Kleitman, who gave the minimum number of comparable pairs appearing in families of a given size. In this talk we shall consider a complimentary problem posed by Erd/H{o}s and Daykin and Frankl in the early ‘80s. They asked for the maximum number of comparable pairs that can appear in a family of $m$ subsets of $[n]$, a quantity we denote by $c(m,n)$. We will first resolve an old conjecture of Alon and Frankl, showing that $c(m,n) = o(m^2)$ when $m = n^{/omega(1)} 2^{n/2}$. Time (and interest) permitting, we shall then discuss some further improvements to bounds on $c(n,m)$ for various ranges of the parameters.

This is joint work with Noga Alon, Roman Glebov and Benny Sudakov.

欢迎感兴趣的师生前来参加！