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   https://doi.org/10.1109/ICASSP40776.2020.9053865  

   Moradipari, Ahmadreza; Alizadeh, Mahnoosh; Thrampoulidis, Christos (May 2020, ICASSP 2020)

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2. Supersaturation of even linear cycles in linear hypergraphs

   https://doi.org/10.1017/S0963548320000206  

   Jiang, Tao; Yepremyan, Liana (September 2020, Combinatorics, Probability and Computing)

   Abstract A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in ann-vertex graph not containingC_2k, the cycle of length 2k, isO(n^1+1/k). Simonovits established a corresponding supersaturation result forC_2k’s, showing that there exist positive constantsC,cdepending only onksuch that everyn-vertex graphGwithe(G)⩾Cn^1+1/kcontains at leastc(e(G)/v(G))^2kcopies ofC_2k, 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 ofr-uniform linear cycles of even length inr-uniform linear hypergraphs. Our proof is self-contained and includes ther= 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. 

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