Determine the Brown–Erdős–Sós Turán number: max edges with no $k$ vertices spanning $s$ edges (Erdős #1157)

Let $r,k,s\ge 2$ with $k>r$. Let $\mathcal{F}$ be the family of all $r$-uniform hypergraphs having exactly $k$ vertices and $s$ edges. Determine the Turán number $$\mathrm{ex}_r(n,\mathcal{F}),$$ the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices that contains no member of $\mathcal{F}$ — equivalently, in which no $k$ vertices span $s$ or more edges.

Acceptance. FULLY RESOLVES: determine the asymptotic order of $\mathrm{ex}_r(n,\mathcal{F})$ for all $r,k,s$ — i.e. the exponent $\theta=\theta(r,k,s)$ with $\mathrm{ex}_r(n,\mathcal{F})=n^{\theta+o(1)}$ — with proof; OR prove the Brown–Erdős–Sós conjecture $\mathrm{ex}_t(n,\mathcal{F})=o(n^t)$ for a case beyond the degenerate $t=2$ ($o(n^2)$) regime already resolved by Shangguan — i.e. for some $t\ge 3$ within the range $k\ge (r-t)s+t+1$. ADVANCES (each independently checkable): resolve a new specific case, pinning the exponent for a triple $(r,k,s)$ where currently only bounds are known, with proof; OR improve the general lower bound $n^{(rs-k)/(s-1)}$ for some parameter family, strictly beyond the bound stated in the background; OR improve the best-known upper bound for a named case with a reproducible argument; OR compute exact values $\mathrm{ex}_r(n;k,s)$ for small $n$ (ILP/SAT) with an optimality certificate, thereby establishing or refuting a conjectured leading exponent. Every improvement must be anchored to 'strictly better than the best bound stated in the background'. Deliver the proof, the improved-bound argument, or the solver output plus optimality certificate.