SCHWEITZER 2010 Problem 3

Let $A_i,i=1,2,\dots,t$ be distinct subsets of the base set $\{1,2,\dots,n\}$ complying to the following condition $$A_ {i} \cap A_ {k} \subseteq A_ {j}$$ for any $1 \leq i <j <k \leq t.$ Find the maximum value of $t.$ Thanks @dgrozev