Let $n>1$ be a natural number. Let $U=\{1,2,...,n\}$, and define $A\Delta B$ to be the set of all those elements of $U$ which belong to exactly one of $A$ and $B$. Show that $|\mathcal{F}|\le 2^{n-1}$, where $\mathcal{F}$ is a collection of subsets of $U$ such that for any two distinct elements of $A,B$ of $\mathcal{F}$ we have $|A\Delta B|\ge 2$. Also find all such collections $\mathcal{F}$ for which the maximum is attained.
1. **Base Case:**
   - Consider \( n = 2 \). The set \( U_2 = \{1, 2\} \).
   - The possible subsets of \( U_2 \) are \( \emptyset, \{1\}, \{2\}, \{1, 2\} \).
   - We need to find collections \( \mathcal{F}_2 \) such that for any two distinct elements \( A, B \in \mathcal{F}_2 \), \( |A \Delta B| \geq 2 \).
   - The only collections satisfying this are \( \mathcal{F}_2 = \{\emptyset, \{1, 2\}\} \) or \( \mathcal{F}_2 = \{\{1\}, \{2\}\} \).
   - In both cases, \( |\mathcal{F}_2| = 2 \leq 2^{2-1} = 2 \).
   - Thus, the base case holds.

2. **Inductive Step:**
   - Assume the statement holds for \( n = k \), i.e., \( |\mathcal{F}_k| \leq 2^{k-1} \) for any collection \( \mathcal{F}_k \subset \mathcal{P}(U_k) \) with the given property.
   - Consider \( n = k+1 \). Let \( U_{k+1} = \{1, 2, \ldots, k, k+1\} \).
   - Define:
     \[
     \mathcal{F}' = \{F \mid k+1 \not\in F, F \in \mathcal{F}_{k+1}\}
     \]
     \[
     \mathcal{F}'' = \{F \mid k+1 \in F, F \in \mathcal{F}_{k+1}\}
     \]
     \[
     \mathcal{F}''' = \{F \setminus \{k+1\} \mid F \in \mathcal{F}''\}
     \]
   - Clearly, \( \mathcal{F}' \cup \mathcal{F}'' = \mathcal{F}_{k+1} \) and \( \mathcal{F}' \cap \mathcal{F}'' = \emptyset \).
   - Thus, \( |\mathcal{F}_{k+1}| = |\mathcal{F}'| + |\mathcal{F}''| = |\mathcal{F}'| + |\mathcal{F}'''| \).
   - Note that \( \mathcal{F}' \) and \( \mathcal{F}''' \) are subsets of \( \mathcal{P}(U_k) \) and also satisfy the given property.
   - By the induction hypothesis, \( |\mathcal{F}'| \leq 2^{k-1} \) and \( |\mathcal{F}'''| \leq 2^{k-1} \).
   - Therefore, \( |\mathcal{F}_{k+1}| \leq 2^{k-1} + 2^{k-1} = 2^k \).

3. **Conclusion:**
   - By induction, \( |\mathcal{F}_n| \leq 2^{n-1} \) for any \( n > 1 \).

4. **Maximal Families:**
   - For \( n = 2 \), the maximal families are \( \{\emptyset, \{1, 2\}\} \) and \( \{\{1\}, \{2\}\} \).
   - For \( n > 2 \), the structure of maximal families can be established via induction.
   - If \( \mathcal{F}_n \) is maximal, then \( \mathcal{F}' \) and \( \mathcal{F}''' \) must also be maximal.
   - They cannot be of the same type (even or odd cardinality), so they must be of different types.
   - Thus, \( \mathcal{F}_n \) will be of the type of \( \mathcal{F}' \).

\(\blacksquare\)

The final answer is \( \boxed{ |\mathcal{F}| \leq 2^{n-1} } \) and the maximal collections are those where subsets have either all even or all odd cardinalities.