Caveat: This is just a random thought, and I realise the question may not be well-posed.
I'm trying to figure out how fast a certain sequence of graphs would grow. Here, I use $n$ as the number of dimensions. The property (let's just call it $P$) the graphs must have is that all the nodes must be connected, and no 2 edges can intersect at any point.
Thus, for n = 1, the largest graph we can make with this property is quite trivial - it is just the graph consisting of two nodes with one edge between them.
Consider another example. For n = 2, the largest graph one can make is $K_4$, which can be embedded as e.g. the vertices and centroid of an equilateral triangle with the edges drawn as straight lines.
Let $f(n)$ be the function that returns the number of nodes the largest $n$-dimensional graph satisfying $P$ would have. So $f(1) = 2$, and $f(2)= 4$.
What would $f(3)$ be and what would the graph look like? I originally thought $f(3) = 6$, where the octahedron graph would be the largest graph satisfying $P$, but later I realised one could just add an extra node outside the tetrahedron, and the graph of those 7 nodes would satisfy $P$. Upon further reflection, I realised that in each successive dimension, there is "more room" to move around, and so as $n$ increases $f(n)$ would start to become very large.
Thus, what would the sequence $f(1), f(2), f(3), f(4)...$ look like, what would the corresponding graphs look like (is there $n$ such that the corresponding graph is not unique i.e. there are multiple graphs that satisfy $P$ and have the maximum number of vertices), and how fast does this sequence grow?
