Distinct volume subsets

Suppose that $a$ and $d$ are positive integers with $a\ge 2$. Let $h_{a,d}(n)$ be the largest integer $t$ such that any set of $n$ points in ${\mathbb{R}}^d$ contains a subset of $t$ points for which all the non-zero volumes of the $\left(\genfrac{}{}{0px}{}{t}{a}\right)$ subsets of order $a$ are distinct. Beginning with Erd\H{o}s in 1957, the function $h_{2,d}(n)$ has been closely studied and is known to be at least a power of $n$. We improve the best known bound for $h_{2,d}(n)$ and show that $h_{a,d}(n)$ is at least a power of $n$ for all $a$ and $d$.