论文标题
超越破碎的四面体
Beyond the broken tetrahedron
论文作者
论文摘要
在这里，我们考虑了Erdős和Sós所建议的均匀密集超图中的HypergraphTurán问题。 Given a $3$-graph $F$, the uniform Turán density $π_u(F)$ of $F$ is defined as the supremum over all $d\in[0,1]$ for which there is an $F$-free uniformly $d$-dense $3$-graph, where uniformly $d$-dense means that every linearly sized subhypergraph has density at least $d$.最近，Glebov，Král'和Volec，以及独立的Reiher，Rödl和Schacht证明了$π_U（K_4^{（3） - }）= \ frac {1} {1} {4} {4} $，解决Erdős和Sós的猜想。尽管非常关注，但统一的Turán密度仍然仅以极少的超图表而闻名。特别是，由于erdős和sós确定$π_U（k_4^{（3）}）$引起的问题保持开放。 在这项工作中，我们通过添加一个额外的顶点，其链接在$ k_4^{（3） - } $上形成匹配，从而从$ k_4^{（3） - } $中确定了五个顶点的均匀图Turán密度，这些顶点是从$ k_4^{（3） - } $获得的。此外，我们指出了确定$π_U（k_4^{（3）}）$的两个天然中间问题，并解决了其中的第一个。
Here we consider the hypergraph Turán problem in uniformly dense hypergraphs as was suggested by Erdős and Sós. Given a $3$-graph $F$, the uniform Turán density $π_u(F)$ of $F$ is defined as the supremum over all $d\in[0,1]$ for which there is an $F$-free uniformly $d$-dense $3$-graph, where uniformly $d$-dense means that every linearly sized subhypergraph has density at least $d$. Recently, Glebov, Král', and Volec and, independently, Reiher, Rödl, and Schacht proved that $π_u(K_4^{(3)-})=\frac{1}{4}$, solving a conjecture by Erdős and Sós. Despite substantial attention, the uniform Turán density is still only known for very few hypergraphs. In particular, the problem due to Erdős and Sós to determine $π_u(K_4^{(3)})$ remains wide open. In this work, we determine the uniform Turán density of the $3$-graph on five vertices that is obtained from $K_4^{(3)-}$ by adding an additional vertex whose link forms a matching on the vertices of $K_4^{(3)-}$. Further, we point to two natural intermediate problems on the way to determining $π_u(K_4^{(3)})$, and solve the first of these.
