Induced even cycles in locally sparse graphs

A graph $G$ is $(c,t)$-sparse if for every pair of vertex subsets $A,B\subset V(G)$ with $|A|,|B|\geq t$, $e(A,B)\leq (1-c)|A||B|$. In this paper we prove that for every $c>0$ and integer $\ell$, there exists $C>1$ such that if an $n$-vertex graph $G$ is $(c,t)$-sparse for some $t$, and has at least $C t^{1-1/\ell}n^{1+1/\ell}$ edges, then $G$ contains an induced copy of $C_{2\ell}$. This resolves a conjecture of Fox, Nenadov and Pham.