14 problems

Let $\mathcal{F}$ be a finite family of axis-parallel rectangles. Write $\tau$ for the minimum number of points meeting every member of $\mathcal{F}$, and $\nu$ for the maximum num…

For a $d$-polytope, let the transversal ratios be denoted by $\rho_d^\mathrm{P}$ and $\rho_d^\mathrm{S}$, corresponding to polytopes and simplicial polytopes, respectively. A polyt…

Transversal $(p,q)$-conjecture. For every $p,q,d$ and $\rho$ there exists a constant $C$ such that, if among any $p$ members of $4\mathcal{F}4$ there are $k+2$ members with a commo…

Let $n\ge 2$, and let $\mathcal{F}_1,\mathcal{F}_2,\dots,\mathcal{F}_n$ be families of translates of a convex compact set $\mathcal{K}$ in the plane. Assume that … for every…

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

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…

Let $\mathcal{F}$ and $\mathcal{G}$ be finite families of convex sets in $\mathbb{R}^3$ such that … for every $A\in\mathcal{F}$ and $B\in\mathcal{G}$. A line transversal to a famil…

Let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^3$, with every two elements of $\mathcal{F}$ intersecting. A line transversal to a subfamily is a line intersecti…

Let $\mathcal{F}$ be an intersecting family of convex sets in $\mathbb{R}^3$. For each triple $T=\{F_1,F_2,F_3\}\in {\mathcal{F}\choose 3}$, let $D_T\subset S^2$ consist of the dir…

Transversal-dimension conjecture. For every integer $k$ with $1\leq k\leq d$, there exist numbers $h(k,d)$ such that, for every $d$-colored family $\mathcal{F}$ satisfying…

Let ${{\mathscr F}}$ be a collection of closed, convex sets in the plane with the $(4,3)$-property, meaning that among every four members, some three have a point in common, and su…

Let ${{\mathscr F}}$ be a collection of closed, convex sets in the plane with the $(4,3)$-property, meaning that among every four members, some three have a point in common, and su…

Let $S_1,\ldots,S_n$ be a scrambled system of bases of $\mathbb{R}^n$: each $S_i$ has size $n$, and $\bigcup S_i$ is the multiset union of $n$ bases. An independent transversal con…

A scrambled system of bases is a set $S_1,\ldots,S_n$ of subsets of $\mathbb{R}^n$, each of size $n$, such that $\bigcup S_i$ is the multiset union of $n$ bases; an independent tra…