VC-dimension sunflower conjecture

Let $f^d_r(k)$ be the least positive integer $m$ such that every family $\mathcal F$ of $k$-sets with $|\mathcal F|\geq m$ and $\text{VC-dim}(\mathcal F)\leq d$ contains an $r$-sunflower. VC-dimension sunflower conjecture. For $d\geq 1$ and $r\geq 3$, there is a constant $C=C(d,r)$ such that

$$f^d_r(k) \leq C^k.$$

This is a weaker form of the Erdős–Rado sunflower conjecture, specialized to families of bounded Vapnik–Chervonenkis dimension. The paper states that it remains open even under the bounded VC-dimension assumption.