Let $p$ be an odd prime number. Let $S=a_1,a_2,\dots$ be the sequence defined as follows: $a_1=1,a_2=2,\dots,a_{p-1}=p-1$, and for $n\ge p$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that in $a_1,a_2,\dots,a_n$ there are no arithmetic progressions of length $p$. We say that a positive integer is a [i]ghost[/i] if it doesn’t appear in $S$.
What is the smallest ghost that is not a multiple of $p$?

[i]Proposed by Guerrero[/i]
1. **Define the sequence \( S \)**:
   - The sequence \( S = a_1, a_2, \ldots \) is defined such that \( a_1 = 1, a_2 = 2, \ldots, a_{p-1} = p-1 \).
   - For \( n \ge p \), \( a_n \) is the smallest integer greater than \( a_{n-1} \) such that in \( a_1, a_2, \ldots, a_n \) there are no arithmetic progressions of length \( p \).

2. **Identify the concept of a ghost**:
   - A positive integer is called a *ghost* if it does not appear in the sequence \( S \).

3. **Analyze the sequence for \( n \ge p \)**:
   - We need to ensure that adding \( a_n \) to the sequence does not create an arithmetic progression of length \( p \).

4. **Consider the case when \( n \) is not a multiple of \( p \)**:
   - If \( p \nmid n \), then adding \( n \) to the sequence does not create an arithmetic progression of length \( p \). This is because any arithmetic progression of length \( p \) must contain a multiple of \( p \), and since \( n \) is not a multiple of \( p \), it cannot complete such a progression.

5. **Consider the case when \( n \) is a multiple of \( p \)**:
   - If \( n = kp \) for some integer \( k \), then adding \( n \) to the sequence would create an arithmetic progression of length \( p \). Specifically, the numbers \( kp - p + 1, kp - p + 2, \ldots, kp \) would form an arithmetic progression of length \( p \).
   - Therefore, we must skip this multiple of \( p \) and make it a ghost.

6. **Conclusion**:
   - From the above analysis, we see that all ghosts are multiples of \( p \).
   - Therefore, there is no ghost which is not a multiple of \( p \).

The final answer is \(\boxed{\text{There is no ghost which is not a multiple of } p}\).