To quantify the richness of a collection C of sets, we use the concept of shattering coefficients (also known as the growth function).
S_C(n) = \max_{x_1,x_2,\dots,x_n \in \Omega } \operatorname{card} \{\,\{\,x_1,x_2,\dots,x_n\}\cap s, s\in C \} .