Unit-area triangles: how many triangles of the same area can $n$ planar points span? (Erdős #1086)

Let $g(n)$ be minimal such that any set of $n$ points in $\mathbb{R}^2$ contains the vertices of at most $g(n)$ triangles with the same area — equivalently, how many triangles of area exactly $1$ can a set of $n$ points in $\mathbb{R}^2$ determine? Estimate $g(n)$. More generally, let $g_d^{r}(n)$ be minimal such that any set of $n$ points in $\mathbb{R}^d$ contains the vertices of at most $g_d^{r}(n)$ $r$-dimensional simplices with the same $r$-dimensional volume; estimate $g_d^r(n)$.

Acceptance. FULLY RESOLVES: a complete proof (machine-checkable Lean/Coq preferred, else a full written proof) determining the order of growth of $g(n)$ up to $n^{o(1)}$ — i.e. matching upper and lower bounds closing the $n^2\log\log n$ vs $n^{20/9}$ gap. ADVANCES: (a) a proven upper bound with exponent strictly below $20/9$, or a construction with proven count asymptotically exceeding $n^2\log\log n$ — in each case strictly better than the best bound stated in the background; (b) in higher dimensions, an improvement of the $g_3^2(n)\ll n^{2.4286}$ bound, of Purdy's $n^{3-c}$, or a proof/refutation of the Erdős–Purdy conjecture that the Lenz-type bound $g_{2k+2}^k(n)\asymp n^{k+1}$ is tight (any single $k$, e.g. $g_4^1$-adjacent cases, qualifies); (c) certified exact values of $g(n)$ for small $n$ (real-algebraic or exhaustive-over-combinatorial-types computation with an optimality certificate), or an explicitly parameterized family with reproducibly computed same-area counts that sharpens the known lower-bound constant. Deliver the proof file, or the construction/values with counting code and verification certificates.