On the asymptotic behavior of the classical Turán numbers

Abstract: A subset of vertices in a hypergraph H is called independent if it does not contain an edge of $H$. The independence number $\alpha(H)$ is the size of the largest independent set. The classical Turán number $T(n,\alpha+1,r)$ is the minimum number of edges in an $n$-vertex $r$-uniform hypergraph $H$ with $\alpha(H) \le \alpha$. In other words, $\binom{n}{r} - T(n,k,r)$ is the largest number of edges in an $n$-vertex $r$-uniform hypergraph that does not contain a complete k-vertex subgraph.

The limit of $T(n,k,r) / \binom{n}{r}$ with $n\to\infty$ is known as Turán density $t(k,r)$. Pál Turán in 1941 proved that $t(\alpha+1,2) = 1/\alpha$. It is widely believed that $t(\alpha+1,3) = 4/\alpha^2$. I will discuss the asymptotic behavior of $t(k,r)$ in respect to $k$ and $r$. I will also cover similar topics for the codegree Turán problem.

combinatoricsprobability

Audience: researchers in the topic

Extremal and probabilistic combinatorics webinar

Series comments: We've added a password: concatenate the 6 first prime numbers (hence obtaining an 8-digit password).