Let the number of floors be n.

For each floor F, each elevator that stops at F stops at only 5 other floors, and thus connects 5 pairs that include F. If n is more than 11, then for each floor F there are more than 10 pairs that include F, and thus more than 2 elevators must stop at F.

Each elevator stops at 6 floors, so the 7 elevators colletively stop at 7 * 6 = 42 floors. For each floor F, more than 2 elevators must stop at F, thus there cannot be more than 42/3 = 14 floors.

Example:

Elevator 1 stops at floors 1 2 3
Elevator 2 stops at floors 1 4 5
Elevator 3 stops at floors 1 6 7
Elevator 4 stops at floors 2 4 6
Elevator 5 stops at floors 2 5 7
Elevator 6 stops at floors 3 4 7
Elevator 7 stops at floors 3 5 6

and in addition any elevator that stops at floor n also stops at floor n+7.

The 7 sets of 3 elements from {1, 2, 3, 4, 5, 6, 7} are the 7 lines of the Fano plane.