**Speaker**: Yinfeng Zhu (祝隐峰), Shanghai Jiao Tong University

**Date**: Fri, May 24, 2019

**Time**: 10:45 - 11:15

**Venue**: Room 630

**Title**: Up and down in a tree

**Abstract**:

Let $(V,2^V,P)$ be a discrete probability space. Consider a rooted tree $T$ with vertex set $V$ and root $r$. A subset $D$ of $V$ is called $T$-down if $d\in D$ implies that all the vertices which do not lie in the same component of $T-d$ with $r$ fall into $D$. For every positive integer $k$, we show that there is either a set $U$ such that $P(U)\geq \frac{1}{2k-1}$ and $U$ induces a path with $r$ as one endpoint, or there are $k$ disjoint $T$-down subsets $D_1,\ldots,D_k$ such that $P(D_1)\geq \cdots \geq P(D_k)\geq \frac{1}{2k-1}$. Related results and questions will also be discussed.

Ongoing joint work with Yaokun Wu.