Rainbow sunflower conjecture

Let $X$ be a finite ground set and let $\mathcal{F}$ be a $w$-set system, meaning that every member of $\mathcal{F}$ has size at most $w$. Color each element of $X$ independently and uniformly red, green, or blue. Rainbow sunflower conjecture. There exists a constant $C$ such that, whenever $|\mathcal{F}|\geq C^w$, with high probability there are distinct sets $S_i,S_j,S_k\in\mathcal{F}$ such that, writing

$$Y=S_i\cap S_j\cap S_k,$$

all elements of $S_i\setminus Y$ are red, all elements of $S_j\setminus Y$ are green, and all elements of $S_k\setminus Y$ are blue.

The conjecture is a colored, adaptive strengthening of the three-petal sunflower problem. The paper proves it with the threshold $C^w$ replaced by $(\log w)^{w(1+o(1))}$, and Rao improves this to $(C\log w)^w$; the constant-exponential form remains open in the source.