I am competing in a programming contest where the submission phase can be stated abstractly as follows : There is a known universe set, $U$, and a hidden target $T \subset U$. I submit $S \subset U$, and for feedback I am given $|S \cap T|$.
What is a good/optimal strategy for finding $T$ in as few submissions as possible given no assumptions on the distribution of $T$? What are the complexity bounds in terms of $|U|$ and $|T|$? In my application, $|U|$ is in the tens of millions and $|T|$ has a known size of around 5000.