Estimate $m(n,k)$, or even better, give an asymptotic formula.
Estimate $m(n,k)$, or even better, give an asymptotic formula.
This is sometimes known as the weak sunflower problem (see [20] for the strong sunflower problem).
When $k=3$ this is strongly connected to the cap set problem (finding the maximal size of subsets of $\mathbb{F}_3^n$ with no three-term arithmetic progressions), as observed by Alon, Shpilka, and Umans [ASU13]). Naslund and Sawin [NaSa17] have proved that \[m(n,3) \leq (3/2^{2/3})^{(1+o(1))n}.\]