In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).
1. **Define the problem and variables:**
   - Let \( n \) be the number of teams, with \( n \ge 4 \).
   - Each pair of teams meets exactly once.
   - The final scores form an arithmetic sequence where each team scores 1 more point than the following team on the scoreboard.
   - Let \( a \) be the score of the lowest scoring team.

2. **Sum of the arithmetic sequence:**
   - The scores form an arithmetic sequence: \( a, a+1, a+2, \ldots, a+(n-1) \).
   - The sum of the scores of all teams is:
     \[
     S = a + (a+1) + (a+2) + \cdots + (a+(n-1))
     \]
   - This can be simplified using the formula for the sum of an arithmetic series:
     \[
     S = na + \frac{(n-1)n}{2}
     \]

3. **Upper bound of the total score:**
   - In a football tournament, each game results in a total of 3 points (either 3-0, 0-3, or 1-1).
   - There are \(\binom{n}{2} = \frac{n(n-1)}{2}\) games in total.
   - Therefore, the maximum possible total score is:
     \[
     \frac{3n(n-1)}{2}
     \]

4. **Inequality for the sum of scores:**
   - The sum of the scores \( S \) must be less than or equal to the maximum possible total score:
     \[
     na + \frac{n(n-1)}{2} \leq \frac{3n(n-1)}{2}
     \]

5. **Solve for \( a \):**
   - Rearrange the inequality to solve for \( a \):
     \[
     na + \frac{n(n-1)}{2} \leq \frac{3n(n-1)}{2}
     \]
     \[
     na \leq \frac{3n(n-1)}{2} - \frac{n(n-1)}{2}
     \]
     \[
     na \leq \frac{2n(n-1)}{2}
     \]
     \[
     na \leq n(n-1)
     \]
     \[
     a \leq n-1
     \]

6. **Conclusion:**
   - The maximum possible score of the lowest scoring team is \( n-1 \).

The final answer is \( \boxed{ n-2 } \).