AI 中文总结
本文证明了对于所有整数 r,k≥3，不含线性环 C_k^{(r)} 的 r-超图数量为 2^{(1+o(1))ex_r(n,C_k^{(r)})}，完全解决了 Balogh 等人的猜想，采用平衡过饱和与容器方法，并在特殊情形使用多色熵定理。
AI 中文摘要
设 $C_{k}^{(r)}$ 为包含 $k$ 条超边的 $r$ 一致线性环。若一个 $r$-图不含 $C_{k}^{(r)}$ 的副本，则称其为 $C_{k}^{(r)}$-free。令 $\operatorname{ex}_{r}(n,C_{k}^{(r)})$ 表示在 $n$ 个顶点上不含 $C_{k}^{(r)}$ 的 $r$-图的最大超边数。Balogh、Narayanan 和 Skokan 提出如下问题：对于每一对整数 $r,k\ge 3$，在 $n$ 个标号顶点上的 $C_k^{(r)}$-free $r$-图的数量是否为 $2^{(1+o(1))\operatorname{ex}_{r}(n,C_{k}^{(r)})}$。尽管对于图（$r=2$）该类似陈述已知不成立（由 Morris 和 Saxton 的构造），但对于超图的一般问题仍然开放。最近，Jiang 和 Longbrake 在 $r\geq5$ 时肯定地回答了该问题。在本文中，我们通过证明对于每一对整数 $r,k\geq3$ 答案都是肯定的，完全解决了该问题。当 $(r,k)\neq(3,3)$ 时，我们建立线性环的平衡过饱和结果，并将其与超图容器方法结合。对于 $(r,k)=(3,3)$，我们转而根据对码度分解 $3$-图，将包含大码度对的超边所构成的子超图编码为有向图，并应用多色熵定理。
英文摘要
Let $C_{k}^{(r)}$ be the $r$-uniform linear cycle on $k$ hyperedges. An $r$-graph is $C_{k}^{(r)}$-free if it contains no copy of $C_{k}^{(r)}$. Let $\operatorname{ex}_{r}(n,C_{k}^{(r)})$ denote the maximum number of hyperedges in an $n$-vertex $C_{k}^{(r)}$-free $r$-graph. Balogh, Narayanan and Skokan asked whether, for every pair of integers $r,k\ge 3$, the number of $C_k^{(r)}$-free $r$-graphs on $n$ labelled vertices is \[ 2^{(1+o(1))\operatorname{ex}_{r}(n,C_{k}^{(r)})}. \] While the analogous statement is known to fail for graphs ($r=2$), by a construction of Morris and Saxton, the general question remained open for hypergraphs. Very recently, Jiang and Longbrake answered the question affirmatively when $r\geq5$. In this paper, we completely resolve the problem by proving that the answer is affirmative for every pair of integers $r,k\geq3$. When $(r,k)\neq(3,3)$, we establish balanced supersaturation results for linear cycles and combine them with the hypergraph container method. For $(r,k)=(3,3)$, we instead decompose $3$-graphs according to their pair-codegrees, encode the subhypergraph formed by the hyperedges that contain a large codegree pair as a directed graph, and apply a multicolour entropy theorem.
