A tournament is held among $n$ teams, following such rules:
a) every team plays all others once at home and once away.(i.e. double round-robin schedule)
b) each team may participate in several away games in a week(from Sunday to Saturday).
c) there is no away game arrangement for a team, if it has a home game in the same week.
If the tournament finishes in 4 weeks, determine the maximum value of $n$.
1. **Understanding the Problem:**
   - We have \( n \) teams in a double round-robin tournament.
   - Each team plays every other team twice: once at home and once away.
   - Each team can have multiple away games in a week but cannot have an away game if it has a home game in the same week.
   - The tournament must be completed in 4 weeks.

2. **Proving \( n = 7 \) is not possible:**
   - Let \( S_i \) (for \( i = 1, 2, \ldots, 7 \)) denote the set of weeks when team \( i \) plays at home.
   - Since the tournament finishes in 4 weeks, \( S_i \) are nonempty proper subsets of \(\{1, 2, 3, 4\}\).
   - According to rule (c), \( S_i \) is not a subset of any other \( S_j \), otherwise team \( i \) cannot play a home game against team \( j \).

3. **Counting Nonempty Proper Subsets:**
   - The set \(\{1, 2, 3, 4\}\) has \(2^4 = 16\) subsets.
   - Excluding the empty set and the full set \(\{1, 2, 3, 4\}\), we have \(16 - 2 = 14\) nonempty proper subsets.

4. **Dividing Subsets into Groups:**
   - We divide these 14 subsets into six groups:
     \[
     \begin{aligned}
     A &= \{\{1\}, \{1, 2\}, \{1, 2, 3\}\}, \\
     B &= \{\{2\}, \{2, 3\}, \{2, 3, 4\}\}, \\
     C &= \{\{3\}, \{1, 3\}, \{1, 3, 4\}\}, \\
     D &= \{\{4\}, \{1, 4\}, \{1, 2, 4\}\}, \\
     E &= \{\{2, 4\}\}, \\
     F &= \{\{3, 4\}\}.
     \end{aligned}
     \]
   - By the pigeonhole principle, if we have 7 teams, there must be at least two teams \( i \) and \( j \) such that \( S_i \) and \( S_j \) belong to the same group.

5. **Contradiction:**
   - If \( S_i \) and \( S_j \) belong to the same group, then \( S_i \subseteq S_j \) or \( S_j \subseteq S_i \), which contradicts the requirement that no \( S_i \) is a subset of any other \( S_j \).

6. **Maximum \( n \):**
   - Therefore, \( n \) can be at most 6.

7. **Constructing a Valid Schedule for \( n = 6 \):**
   - Let \( S_1 = \{1, 2\} \), \( S_2 = \{1, 3\} \), \( S_3 = \{1, 4\} \), \( S_4 = \{2, 3\} \), \( S_5 = \{2, 4\} \), \( S_6 = \{3, 4\} \).
   - Verify that no \( S_i \) is a subset of any other \( S_j \) and that all teams can play their home games within the 4 weeks.

The final answer is \( \boxed{ n = 6 } \).