Unavoidable Hypergraphs

The following very natural problem was raised by Chung and Erd\H{o}s in the early 80's has since been repeated a number of times. What is minimum Tur\'an $\text{ex}(n,\mathcal{H})$ among all $r$-graphs $\mathcal{H}$ with fixed edges? Their actual focus on an equivalent perhaps even more question which asks what largest size $r$-graph that can not be avoided any $n$ vertices $e$ In original paper they resolve this asymptotically for graphs, most range $e$. follow-up work $3$-uniform case raise $4$-uniform as next step. we make first progress over 40 years resolving gives us some indication how answer should behave general.