A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on k such that every n-vertex graph G with e(G)⩾ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).In this paper we extend Simonovits' result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.

中文翻译：

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线性超图中偶数线性循环的过饱和

Erdős 以及 Bondy 和 Simonovits [3] 的经典结果表明，n-顶点图不包含C2k, 长度为 2 的循环ķ， 是○(n1+1/ķ）。Simonovits 建立了相应的过饱和结果C2ķ, 表明存在正常数C,C只取决于ķ这样每一个n-顶点图G和e(G)⩾cn1+1/ķ至少包含C(e(G)/v(G))2ķ的副本C2ķ，这个副本数是由随机图紧密实现的（高达一个乘法常数）。在本文中，我们将 Simonovits 的结果扩展到过饱和结果r- 均匀长度的线性循环r- 均匀线性超图。我们的证明是独立的，包括r= 2 例。作为辅助工具，我们开发了从一般宿主图到几乎规则宿主图的归约引理，可用于其他过饱和问题，因此可能具有独立的兴趣。