4 problems

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

Let $p$ be a positive integer, and let $\mathcal{F}$ be a family of axis-parallel rectangles in the plane. The family has the $(p+1,2)$ property if among any $p+1$ rectangles some…

Let $\mathcal{F}$ be a family of axis-parallel rectangles in $\mathbb{R}^2$. It satisfies the $(p,2)$-property when among every $p$ rectangles, some $2$ have a non-empty intersecti…

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…