Given $m$ sets $S_1, S_2, \dots, S_m$ and a bound $b$, find as many sets as possible among $m$ sets, says $S_{i_i}, S_{i_2}, \dots, S_{i_k}$ such that

$$\big| S_{i_i} \cup S_{i_2} \cup \cdots \cup S_{i_k} \big| < b$$ I came across the problem above in computer science research. Since my mathematics background is so-so, I would really appreciate any hints.