For every integer ℓ≥2, any (1−ε,t)-sparse n-vertex graph with at least C t^{1−1/ℓ} n^{1+1/ℓ} edges contains at least C' d^{2ℓ} induced copies of the even cycle C_{2ℓ}, where d is its average degree.
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Supersaturation of induced even cycles in locally sparse graphs
For every integer ℓ≥2, any (1−ε,t)-sparse n-vertex graph with at least C t^{1−1/ℓ} n^{1+1/ℓ} edges contains at least C' d^{2ℓ} induced copies of the even cycle C_{2ℓ}, where d is its average degree.