Random Tur\'an Problems for $K_{s,t}$ Expansions

Let $K_{s,t}^{(r)}$ denote the $r$-uniform hypergraph obtained from the graph $K_{s,t}$ by inserting $r-2$ new vertices inside each edge of $K_{s,t}$. We prove essentially tight bounds on the size of a largest $K_{s,t}^{(r)}$-subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$ whenever $r\ge 2s/3+2$, giving the first random Tur\'an results for expansions that go beyond a natural "tight-tree barrier." In addition to this, our methods yield optimal supersaturation results for $K_{s,t}^{(3)}$ for sufficiently dense host hypergraphs, which may be of independent interest.