8 problems

Hadwiger–Debrunner's conjecture. For every $p\geqslant q\geqslant d+1$, there exists a constant $c=c(p,q;d)$ such that every family of compact, convex sets in $\mathbb{R}^d$ with t…

Dol'nikov's conjecture. There exists $j\in[3]$ such that $\mathcal{F}_j$ can be pierced by $3$ points.

Jung–Keszegh–Pálvölgyi–Yuditsky conjecture. For all $k\geq2$ there exists a constant $c_k<1$ such that, whenever

Fractional line-piercing conjecture. There exists a constant $c$ such that some line intersects at least $c|\mathcal{F}|$ members of $\mathcal{F}$.

Projected-family piercing conjecture. There exists a constant $A$ such that every finite family of axis-parallel rectangles satisfies

Let $\mathcal{F}$ be a family of compact, convex sets in the plane, and let $L(k)$ denote the minimum number of lines guaranteed to pierce $\mathcal{F}$ when $\mathcal{F}$ contains…

Generalized Erdős–Grünbaum conjecture. The family should satisfy

Erdős–Grünbaum's conjecture. There is a constant $n$ such that every such family has a piercing set consisting of $n$ points.