The largest complete graph that embeds in 2 dimensions is $K\_4$, while the largest complete graph that embeds in 3 dimensions is $K\_{\infty}$, right? However, I don't know any constructive proof of it.


**Informal Explanation**:
What is the max number of points in $\mathbb{R}^3$, interconnected by lines of any curvature, such that no line intersects any other line? Each point is connected with all other points.
For $\mathbb{R}^2$ it is only 4 points (smth. like Mercedes symbol) - why 4 and not 3 or 5? How many points are possible to connect in such way in $\mathbb{R}^3$? (I suggest, infinite number, but it is interesting to look at a proof). What are some special properties of the Euclidean $\mathbb{R}^3$ such that the number of interconnected points jumps from 4 in $\mathbb{R}^2$ to infinity in $\mathbb{R}^3$?


**PS**: I don't understand why my question has got already 4 downvotes? No comments, no critics, why? English is not my native language, that's why?



What is the max number of points in R^3, interconnected by generic curves?
Take straight lines connecting the points $(t, t^2, t^3), t \in \mathbb{N}$. As far as I can tell you can also boost this to $t \in \mathbb{R}$. The point here is that two distinct lines between points on this curve intersect if and only if the four points involved lie on a plane (or there are only three points involved, but in that case you already know what the intersection is), but any plane $ax + by + cz = d$ intersects the curve in at most three points because $at + bt^2 + ct^3 = d$ has at most three roots.


