Suppose that $n\ge 8$ persons $P_1,P_2,\dots,P_n$ meet at a party. Assume that $P_k$ knows $k+3$ persons for $k=1,2,\dots,n-6$. Further assume that each of $P_{n-5},P_{n-4},P_{n-3}$ knows $n-2$ persons, and each of $P_{n-2},P_{n-1},P_n$ knows $n-1$ persons. Find all integers $n\ge 8$ for which this is possible.

(It is understood that "to know" is a symmetric nonreflexive relation: if $P_i$ knows $P_j$ then $P_j$ knows $P_i$; to say that $P_i$ knows $p$ persons means: knows $p$ persons other than herself/himself.)
1. **Initial Setup and Assumptions:**
   - We have \( n \geq 8 \) persons \( P_1, P_2, \ldots, P_n \).
   - For \( k = 1, 2, \ldots, n-6 \), person \( P_k \) knows \( k+3 \) persons.
   - Each of \( P_{n-5}, P_{n-4}, P_{n-3} \) knows \( n-2 \) persons.
   - Each of \( P_{n-2}, P_{n-1}, P_n \) knows \( n-1 \) persons.

2. **Reduction of the Problem:**
   - Since \( P_{n-2}, P_{n-1}, P_n \) know everyone else, we can remove them from consideration.
   - Denote \( P_{n-5} = X \), \( P_{n-4} = Y \), \( P_{n-3} = Z \).
   - The degrees of the remaining vertices are \( 1, 2, \ldots, n-6, n-5, n-5, n-5 \).

3. **Analysis of Connections:**
   - \( P_1 \) knows \( 1+3 = 4 \) persons, so \( P_1 \) does not know at least \( n-5-4 = n-9 \) persons among \( X, Y, Z \).
   - \( P_2 \) knows \( 2+3 = 5 \) persons, so \( P_2 \) does not know at least \( n-5-5 = n-10 \) persons among \( X, Y, Z \).

4. **Specific Cases:**
   - For \( n = 8 \):
     - \( P_1 \) knows 4 persons, \( P_2 \) knows 5 persons.
     - \( P_3 \) knows 6 persons, and so on.
     - \( P_1 \) and \( P_2 \) can be arranged to not know \( X, Y, Z \) in a way that satisfies the conditions.
   - For \( n = 9 \):
     - \( P_1 \) knows 4 persons, \( P_2 \) knows 5 persons.
     - \( P_3 \) knows 6 persons, and so on.
     - Similar to \( n = 8 \), \( P_1 \) and \( P_2 \) can be arranged to not know \( X, Y, Z \) in a way that satisfies the conditions.

5. **Contradiction for \( n \geq 10 \):**
   - For \( n \geq 10 \), \( P_1 \) and \( P_2 \) not knowing \( X, Y, Z \) leads to a contradiction.
   - Specifically, \( P_3 \) knowing \( X, Y, Z \) implies \( P_3 \) does not know \( P_{n-6} \).
   - Since \( P_{n-6} \) does not know \( P_1, P_2, P_3 \), the degree of \( P_{n-6} \) cannot be \( n-6 \), leading to a contradiction.

6. **Verification for \( n = 8 \) and \( n = 9 \):**
   - For \( n = 8 \) and \( n = 9 \), the conditions can be satisfied without contradictions.

The final answer is \( \boxed{ n = 8, 9 } \).