3 problems

Colorful four-point piercing conjecture. Either $\mathcal{F}_{1}$ or $\mathcal{F}_{2}$ can be pierced by $4$ points.

Dol’nikov's generalized conjecture. There exists $j \in \{1,2,\dots,n\}$ such that $\bigcup_{i \neq j} \mathcal{F}_{i}$ can be pierced by $3$ points.

Let $p$ and $q$ be integers with $p\geq q\geq 3$, let $S$ be a surface, and let $\mathcal{F}$ be a finite family of open connected subsets of $S$. The $(p,q)$ property means that a…