10 problems

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…

Fractional transversal conjecture. There is a function $f$ such that

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.

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 $\mathcal{F}$ 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 $\mathcal{F}$ 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 $S_1,\ldots,S_n$ 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…