You might have some info in David Conlon's lecture 14, taking about Moon-moser inequality for hypergraph, due to Caen (alas I can't access Caen paper) its.caltech.edu/~dconlon/Extremal-course.html

Google translate gives a pretty good translation of the pdf document. you would have to add the math statement yourself but it's doable. Note that the v2 seems to include the comment made by Vidali at the end of the youtube video.

@FedorPetrov yep that's my guess. I know the proof, I could write a summary of it. I was just hoping for some reference. Thanks

Thanks @LouisD, I managed to track down the actual result, not trivial, not that difficult.

Hi Brendan, I could restrict to $p\sim n^{1-h+1/\ell}$. So a linear hypergraph would have at most $\frac{n(n-1)}{h(h-1)}$ edges while the random hypergraph has $\binom{n}{h}p \sim n^{1+1/\ell}\cdot h^{-h}$ edges (I think), so $p$ should be small enough.

you could event add the edge $(sv)$ making your graph weakly connected without containing the desired path. On a side note, I don't see what does the condition $(st)\in\mathcal{A}_G$ do. A path from $s$ to $t$ through another vertex $u$ will never use this edge.