5 problems

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

Wegner's conjecture. Every finite family of axis-parallel rectangles in $\mathbb{R}^2$ 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…