ILL 1986 Problem 61

Given a positive integer $n$ , find the greatest integer $p$ with the property that for any function $f : \mathbb P(X) \to C$ , where $X$ and $C$ are sets of cardinality $n$ and $p$ , respectively, there exist two distinct sets $A,B \in \mathbb P(X)$ such that $f(A) = f(B) = f(A \cup B)$ . ( $\mathbb P(X)$ is the family of all subsets of $X$ .)