Must every graph with $n$ vertices and $\delta n^2$ edges contain large subgraphs in which every two edges lie on specified short cycles? A dense high-girth construction refutes the statement when $\delta$ may shrink with $n$.

Must every graph with $n$ vertices and $\delta n^2$ edges contain large subgraphs in which every two edges lie on specified short cycles? A dense high-girth construction refutes the statement when $\delta$ may shrink with $n$.