### On the upper bound for the Tur\'an density of K^r_{r+1}

- Series
- Combinatorics Seminar
- Time
- Friday, September 12, 2008 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Yi Zhao – Georgia State University

Let K^r_{r+1} denote the complete r-graph on r+1 vertices. The Turan density of K^r_{r+1} is the largest number t such that there are infinitely many K^r_{r+1}-free r-graphs with edge density t-o(1). Determining t(K^r_{r+1}) for r > 2 is a famous open problem of Turan. The best upper bound for even r, t(K^r_{r+1})\leq 1-1/r, was given by de Caen and Sidorenko. In a joint work with Linyuan Lu, we slightly improve it. For example, we show that t(K^r_{r+1})\leq 1 - 1/r - 1/(2r^3) for r=4 mod 6. One of our lemmas also leads to an exact result for hypergraphs. Given r > 2, let p be the smallest prime factor of r-1. Every r-graph on n > r(p-1) vertices such that every r+1 vertices contain 0 or r edges must be empty or a complete star.