Let $ A_i,i=1,2,\dots,t$ be distinct subsets of the base set $\{1,2,\dots,n\}$ complying to the following condition
$$ \displaystyle A_ {i} \cap A_ {k} \subseteq A_ {j}$$for any $1 \leq i <j <k \leq t.$ Find the maximum value of $t.$
Thanks @dgrozev
1. **Understanding the Problem:**
   We are given distinct subsets \( A_i \) of the base set \(\{1, 2, \dots, n\}\) such that for any \(1 \leq i < j < k \leq t\), the intersection \( A_i \cap A_k \subseteq A_j \). We need to find the maximum value of \( t \).

2. **Analyzing the Condition:**
   The condition \( A_i \cap A_k \subseteq A_j \) implies that any element \( x \) that is in both \( A_i \) and \( A_k \) must also be in \( A_j \). This suggests that if \( x \) is in \( A_i \) and \( A_j \) for \( i < j \), then \( x \) must be in all sets \( A_s \) for \( i < s < j \).

3. **Intervals of Indices:**
   For each element \( x \) in the base set, we can consider a continuous interval \( I_x \) of indices such that \( x \) belongs to \( A_i \). This interval \( I_x \) is defined as the set of indices \( i \) for which \( x \in A_i \).

4. **Symmetric Difference:**
   For each \( i \leq t-1 \), we can associate some element \( x \) to \( i \) such that \( x \) belongs to the symmetric difference \( A_i \Delta A_{i+1} \). This means \( x \) is in exactly one of \( A_i \) or \( A_{i+1} \), but not both.

5. **Endpoints of Intervals:**
   Each \( x \) is associated with at most 2 indices: the start or end of its interval \( I_x \). Specifically, if \( x \) is in \( A_1 \) or \( A_t \), it is associated with at most one index (either the start or end of \( I_x \)).

6. **Double Counting Argument:**
   We count the pairs \((i, x)\) where \( x \) is associated with \( i \). Since each \( x \) can be associated with at most 2 indices, and there are \( n \) elements in the base set, the total number of such pairs is at most \( 2n \).

7. **Inequality:**
   Since there are \( t-1 \) indices \( i \) (from 1 to \( t-1 \)), and each index can be associated with at least one element \( x \), we have:
   \[
   t-1 \leq 2n
   \]
   Simplifying, we get:
   \[
   t \leq 2n + 1
   \]

8. **Conclusion:**
   The maximum value of \( t \) is \( 2n + 1 \).

The final answer is \( \boxed{2n + 1} \).