I am trying to find a proof or (more likely) counterexample of the following problem:
Given a list of same sized unique sets produce a list of 2-tuple counts contained within.
Using the 2-tuple counts it is possible to reproduce the original sets?
For example...
The following same sized unique sets:
{1, 2, 3, 6} {1, 2, 4, 5} {1, 3, 4, 6} {2, 3, 5, 6} {2, 4, 5, 6} {3, 4, 5, 6}
Contains these 2-tuple counts:
(1, 2) = 2 (2 sets contains both 1 & 2) (1, 3) = 2 (2 sets contains both 1 & 3) (1, 4) = 2 (etc) (1, 5) = 1 (1, 6) = 2 (2, 3) = 2 (2, 4) = 2 (2, 5) = 3 (2, 6) = 3 (3, 4) = 2 (3, 5) = 2 (3, 6) = 4 (4, 5) = 3 (4, 6) = 3 (5, 6) = 3
Using these 2-tuple counts, there is only one possible list of sets which could occur (the original list of sets - found by brute force).
I have no idea where to begin with a proof, and am just trying random combinations hoping to find a counterexample.
Any assistance would be greatly appreciated.
EDIT:
As Petar Ivanov has identified, I am only concerned with sets of a known size (the same as the original).