On Turán numbers of the complete $4$-graphs

Abstract

The Turán 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 $r\ge 2$, there exists $t_{\ast }(r)$ such that $T(n,\alpha +1,r)=t_{\ast }(r)n^r{\alpha }^{1-r}(1+o(1))$ as $\alpha /r\to \infty$ and $n/\alpha \to \infty$. It is known that $t_{\ast }(2)=1/2$, and the conjectured value of $t_{\ast }(3)$ is $2/3$. We prove that $t_{\ast }(4)<0.706335$.

References in corpus (1)

• Rainbow triangles in three-colored graphs