Lower bound of disjointness by discrepancy?
You asked two questions.  I'll answer the second.  A rectangle is a set $R$ of the form

$$R = \{(x_1,\dots,x_n) : \ell_1 \le x_1 \le u_1, \dots, \ell_n \le x_n \le u_n\}$$

for some $\ell_1,\dots,\ell_n,u_1,\dots,u_n$.