On Tur\'{a}n numbers of the complete $4$-graphs

Abstract

The Tur\'{a}n number $T(n,\alpha+1,r)$ is the minimum number of edges in an $n$-vertex $r$-graph whose independence number does not exceed $\alpha$. For each $r\geq 2$, there exists $t_*(r)$ such that $T(n,\alpha+1,r) = t_*(r) \: n^r \: \alpha^{1-r} \: (1+o(1))$ as $\alpha / r \to\infty$ and $n / \alpha \to\infty$. It is known that $t_*(2) = 1/2$, and the conjectured value of $t_*(3)$ is $2/3$. We prove that $t_*(4) < 0.706335\:$.