Find two disjoint set

Question

Given an universum $U$ and two sets $A$ and $B$ of sets of elements from $U$. I want to find (if such a pair exists) $a \in A$ and $b \in B$: $a \cap b \equiv \emptyset$. Currently I can do it only in $O(|A| \cdot |B| \cdot |U|)$, is there way to improve this?

Say, $|U| \leq 32$. Is there a way to speed algorithm up?

If it matters, one can assume, that all elements in $A \cup B$ are unique. Another variation of the problem is $A \equiv B$ and there is a need to search $a$ and $b$ from the single set.

Answer 1

Set $A = B$ and $|U| = \Theta(\log |A|)$, and you run up against the Orthogonal Vectors Conjecture trying to do better than $|A|^{2-o(1)}$.