The Ultimate Board Game

Bengt's friends are playing board games. Bengt, however, is not. He got distracted, as he often does, by trying to improve on what he perceives as flaws in the games. Now he's trying to create a new, better game. As part of that, he needs to design the ultimate board. The board in ths case can be though of as a graph - that is, there are a finite number of "squares", each of which is connected to a finite number of other squares. Connections are symmetric, so if you can go from square P to square Q, you can also go from Q to P. A "region" of the board can be defined as a subset of connected squares, and all the connections between them (a connected induced subgraph, for those into graph theory).

Bengt figures that all regions of at least size k should be different. Otherwise the game gets... repetitive, or something. Given that constraint, he wants to make the board as large as possible.

1. If we know that all regions of size k are unique, does it follow that all larger regions are also unique? Prove / disprove.
2. How many possible size k regions (or boards) are there? Solve for, I don't know, a bunch of different k I guess, maybe find an upper bound of the sequence, or better yet, find the exact function.
3. The real question at hand - how large a board can you make, given that all regions of size k have to be unique? Some k, upper bound, exact function?
4. Boards defined like this can get pretty messy to draw, so we can add an extra rule, that the connections are not allowed to cross - a planar graph, for the graph-theoretically inclined. Again, as in b...
5. ...and as in c.