Consider an $n \times n \times n$ cube. I would like to consider subsets of points in the cube with the two following constraints:
Each row in the cube (in any of the three directions) has exactly 2 or 0 points.
If two points are in the same row (any of three), connect them with an edge. The subset must be connected by these edges.
It is easy to build such a set which is $\Omega(n)$, for instance a staircase, or stacking squares. Is it the case that these structures are also $O(n)$? Or is there an $\Omega(n^2)$ bound?
Building such a set would require adding a linear number of points for each added dimension, which is hard due to the first restriction.
It turns out that these objects have been studied by David Eppstein, and the largest example for an $n \times n \times n$ cube is exactly the maximum $2n^2$.
To see the construction and other comments see:
https://cstheory.stackexchange.com/questions/38462/lower-bound-on-the-largest-restrained-cubic-subset