3 problems

Higher-dimensional sunflower conjecture. For every infinite structure $\mathcal{M}$ and every $k,n\in\mathbb{N}^+$, $\mathcal{M}$ has the infinite $n$-sunflower property in dimensi…

The sunflowerability conjecture for generic $K_n$-free graphs. For all $n\geq 3$, ${\mathcal{H}}_n$ is sunflowerable.

The 2-sunflowerability collapse conjecture. If ${\mathcal{M}}$ is a 2-sunflowerable structure, then ${\mathcal{M}}$ is sunflowerable.