The $r$-uniform linear $k$-cycle $C^r_k$ is the $r$-uniform hypergraph on$k(r-1)$ vertices whose edges are sets of $r$ consecutive vertices in a cyclicordering of the vertex set chosen in such a way that every pair of consecutiveedges share exactly one vertex. Here, we prove a balanced supersaturationresult for linear cycles which we then use in conjunction with the method ofhypergraph containers to show that for any fixed pair of integers $r, k \ge 3$,the number of $C^r_k$-free $r$-uniform hypergraphs on $n$ vertices is$2^{\Theta(n^{r-1})}$, thereby settling a conjecture due to Mubayi and Wang.
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机译：$ r $均匀线性$ k $循环$ C ^ r_k $是$ k $（r-1）$顶点上的$ r $一致超图，其边是在的循环顺序中$ r $个连续顶点的集合。以这样一种方式选择顶点集：每对连续的边精确共享一个顶点。在这里，我们证明了线性循环的平衡过饱和结果，然后结合超图容器的方法使用该结果，表明对于任何固定的整数对$ r，k \ ge 3 $，$ C ^ r_k $ -free $在$ n $个顶点上的r $一致超图是$ 2 ^ {\ Theta（n ^ {r-1}）} $，从而解决了Mubayi和Wang带来的猜想。
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