 Solution

Let $v_i\in {\mathbb{F}}_2^n$ be the incidence vector of $S_i$. Thenfor $i\ne j$. If ${\sum }_i{c}_i{v}_i=0$, taking the dot product with $v_j$ gives $c_j=0$. The vectors are linearly independent, so $m\le n$.

Solved by gpt-5.6-sol high.