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 α. For each $r\ge 2$, 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 $t_{⁎}(4)<0.706335$.

中文翻译：

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关于完全四图的图兰数

图兰数 $吨(n,\alpha +1,r)$是独立数不超过α的n顶点r图中的最小边数。对于每个$r\ge 2$， 那里存在 ${吨}_{⁎}(r)$ 以至于 $吨(n,\alpha +1,r)={吨}_{⁎}(r)n^r{\alpha }^{1-r}(1+○(1))$ 作为 $\alpha /r\to \infty$ 和 $n/\alpha \to \infty$. 众所周知${吨}_{⁎}(2)=1/2$, 的推测值 ${吨}_{⁎}(3)$是 2/3。我们证明${吨}_{⁎}(4)<0.706335$.