论文标题
$ r $ raphs on $ r+1 $顶点的turán数字
Turán numbers of $r$-graphs on $r+1$ vertices
论文作者
论文摘要
令$ h_k^r $表示$ r $ - 均匀的超图和$ k $边缘和$ r+1 $ dertices，其中$ k \ leq r+1 $（很容易看出，这种超图是独特的，直到同构为同构）。全部$ k \ geq 3 $的$π（h_k^r）\ leq \ frac {k-2} $ frac {k-2} $ for $ k \ geq 3 $和$π（h_3^r）\ geq 2^{1-r} $ for $ k = 3 $。我们证明$π（h_k^r）\ geq（c_k-o（1））\，r^{ - （1+ \ frac {1} {k-2}）} $ as $ r \ to \ infty $。在这种情况下，$ k = 3 $，我们证明$π（h_3^r）\ geq（1.7215-o（1））\，r^{ - 2} $ as $ r \ to \ to \ infty $和$π（h_3^r）\ geq r^{ - 2} $ for All $ r $。
Let $H_k^r$ denote an $r$-uniform hypergraph with $k$ edges and $r+1$ vertices, where $k \leq r+1$ (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Turán density are $π(H_k^r) \leq \frac{k-2}{r}$ for all $k \geq 3$, and $π(H_3^r) \geq 2^{1-r}$ for $k=3$. We prove that $π(H_k^r) \geq (C_k - o(1)) \, r^{-(1+\frac{1}{k-2})}$ as $r\to\infty$. In the case $k=3$, we prove $π(H_3^r) \geq (1.7215 - o(1)) \, r^{-2}$ as $r\to\infty$, and $π(H_3^r) \geq r^{-2}$ for all $r$.
