Consider graphs over $n$ nodes.
What is the maximum number of edges of a linklessly embeddable graph?

A more general question is the following. Given $\mu$ what is the maximum number of edges of graphs with the Colin de Verdiere number $\mu$?

A related question would be the following. Is there a fixed space (say a 2-manifold) such that a graph which embeds linklessly into $\mathbb{R}^3$ embeds in that space?

Thanks,