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?