Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 109 / 1 / i / Solution

Associate to each set $A_i\subseteq [n]$ its characteristic vector of a set $v_i\in {\mathbb{F}}_2^n$. The hypotheses sayIf ${\sum }_i{c}_i{v}_i=0$, taking the dot product with $v_j$ gives $c_j=0$. Thus the vectors are linearly independent, so $m\le \dim {\mathbb{F}}_2^n=n$. Equality is attained by the $n$ singleton sets.