论文标题
随机Turán定理用于超图周期
Random Turán theorem for hypergraph cycles
论文作者
论文摘要
给定$ r $ - 均匀的超图$ g $和$ h $turán数字$ \ rm ex（g，h）$是$ h $ free子图中的最大边数。我们研究$ g = g = g = g_ {n，p}^{（r）} $，erdős-rényi随机$ r $ r $ r $ - 均匀的超graph和$ h = c_ {2 \ ell}^{（r）（r）} $，$ r $ r $ r $ liniform-luniform-limatife line $ 2时。图形的情况（$ r = 2 $）是许多研究人员已经调查的长期开放问题。我们确定$ \ rm ex（g_ {n，p}^{（r）}，c_ {2 \ ell}^{（r）}）$ to polylogarithmic因素，除了$ p = p = p（n）$的少数值间隔外，其长度降低为$ \ \ ell。 我们的主要技术贡献是线性甚至循环的平衡过饱和结果，它可以改善Ferber-McKinley-Samotij和Balogh-Narayanan-Skokan的先前结果。新颖的是，过饱和结果取决于基础超图中某些顶点的代码。这种方法可用于证明其他超图$ h $的相似结果。
Given $r$-uniform hypergraphs $G$ and $H$ the Turán number $\rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the Erdős-Rényi random $r$-uniform hypergraph, and $H=C_{2\ell}^{(r)}$, the $r$-uniform linear cycle of length $2\ell$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine $\rm ex(G_{n,p}^{(r)}, C_{2\ell}^{(r)})$ up to polylogarithmic factors for all but a small interval of values of $p=p(n)$ whose length decreases as $\ell$ grows. Our main technical contribution is a balanced supersaturation result for linear even cycles which improves upon previous such results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan. The novelty is that the supersaturation result depends on the codegree of some pairs of vertices in the underlying hypergraph. This approach could be used to prove similar results for other hypergraphs $H$.
