Forbidding a Set Difference of Size 1

How large can a family \cal A \subset \cal P [n] be if it does not contain A,B with |A\setminus B| = 1? Our aim in this paper is to show that any such family has size at most \frac{2+o(1)}{n} \binom {n}{\lfloor n/2\rfloor }. This is tight up to a multiplicative constant of $2$. We also obtain similar results for families \cal A \subset \cal P[n] with |A\setminus B| \neq k, showing that they satisfy |{\mathcal A}| \leq \frac{C_k}{n^k}\binom {n}{\lfloor n/2\rfloor }, where C_k is a constant depending only on k.