Erdős Problem #1075

Let \(r\geq 3\). There exists \(c_r>r^{-r}\) such that, for any \(\epsilon>0\), if \(n\) is sufficiently large, the following holds. Any \(r\)-uniform hypergraph on \(n\) vertices with at least \((1+\epsilon)(n/r)^r\) many edges contains a subgraph on \(m\) vertices with at least \(c_rm^r\) edges, where \(m=m(n)\to \infty\) as \(n\to \infty\).