Finding Least Value m with Property P in Discrete Math

[flashback]

Consider a set X with |X|=n≥1 elements. A family F of distinct subsets of X is sad to have property P if there exist A and B in F, such that A is a proper subset of B and |B\A|=1. Determine the least value m, so that any F with |F|>m has property P.
This is a problem asked by our Discrete mathematics teacher,but i have no idea where to start from.. If anybody could help me?

 

[johng1]

There have been no replies, but the question is solved?

Anyway, the answer is $2^{n-1}$. Here are substantial hints to prove this:

1. There is a collection $\mathcal{F}$ of subsets with |$\mathcal{F}|=2^{n-1}$ that does not satisfy P. One way to prove this is by induction on n.

2. If $\mathcal{F}>2^{n-1}$, $\mathcal{F}$ satisfies P. Think of the subsets as being represented by the binary numbers (bit strings of length n). In a subset of a Gray code with more than $2^{n-1}$ elements, show that the subset contains 2 consecutive elements.