A classical result of Erdős and, independently, of Bondy and Simonovits  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.