On a problem of Duke-Erdos-Rodl on cycle-connected subgraphs

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In this short note, we prove that for β< 1/5 every graph G with n vertices and n²⁻β edges contains a subgraph G' with at least cn²⁻²β edges such that every pair of edges in G' lie together on a cycle of length at most 8. Moreover edges in G' which share a vertex lie together on a cycle of length at most 6. This result is best possible up to the constant factor and settles a conjecture of Duke, Erdos, and Rodl.