http://dcherukhin.i2p/research/research.htm
In particular, for $d=2,3,4$ , our bounds are $\Omega(n^{1.5})$, $\Omega(n \log n)$ and $Omega(n \log \log n)$ respectively;
for $d \ge 5$, the function $\lambda_{d-1}(n)$ is slowly
growing.
On the Boolean model, our bounds are the best known for all even $d$ and for $d=3$ .