$1000
Let $f(n,k)$ be minimal such that every $\mathcal{F}$ family of $n$-uniform sets with $\lvert F\rvert \geq f(n,k)$ contains a $k$-sunflower. Is it true that
\[f(n,k) < c_k^n\]
for some constant $c_k>0$?
Erdős and Rado
[ErRa60] originally proved $f(n,k)\leq (k-1)^nn!$. Kostochka
[Ko97] improved this slightly, but the bound stood at $n^{(1+o(1))n}$ for a long time until Alweiss, Lovett, Wu, and Zhang
[ALWZ20] proved
\[f(n,k) < (Ck\log n\log\log n)^n\]
for some constant $C>1$. This was refined slightly, independently by Rao
[Ra20], Frankston, Kahn, Narayanan and Park
[FKNP19], and Bell, Chueluecha and Warnke
[BCW21], leading to the current record of
\[f(n,k) < (Ck\log n)^n\]
for some constant $C>1$. The usual focus is on the regime where $k=O(1)$ is fixed (say $k=3$) and $n$ is large, although for the opposite regime Kostochka, Rödl, and Talysheva
[KoRoTa99] have shown
\[f(n,k)=(1+O_n(k^{-1/2^n}))k^n.\]