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