##### Department of Mathematics,

University of California San Diego

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### Math 269 - Seminar in Combinatorics

## Yi Zhao

#### Department of Mathematics and Statistics \\ Georgia State University

## An exact result and its application on hypergraph Tur\\'an numbers

##### Abstract:

We first prove an exact result for hypergraphs: given $r\ge 2$, let $p$ be the smallest prime factor of $r-1$. If $n> (p-1)r$ and $G$ is an $r$- uniform hypergraph on $[n]$ such that every $r+1$ vertices contain $0$ or $r$ edges, then $G$ is either empty or a star, $\{E\subset [n]: |E|=r, E\ni x\}$ for some $x\in [n]$. Then we use it to slightly improve best known bounds for hypergraph Tur\'an numbers. We show that $\pi(K^r_{r+1})\leq 1- \frac{1}{r} - \left(1- \frac{1}{r^{p-1}}\right)\frac{(r-1)^2}{2r^p({r+p\choose p-1}+{r+1\choose 2})}$ when $r\ge 4$ is even. This is joint work with Linyuan Lu.

Host: Jeff Remmel

### December 18, 2007

### 3:00 PM

### AP&M 7321

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