3 problems

For each $1\leq i\leq n$, let $f_i$ be a monotone function and let $\alpha_i$ be a specified value. A family of sets has a $(f,\alpha)$-transversal when one transversal satisfies t…

Let $\mathcal{B} = \{B_1,\ldots, B_n\}$ be a finite family of disjoint cylinders in $\mathbb{R}^3$. A set of points $P = \{p_1,\ldots, p_n\}$ spans $\mathbb{R}^k$ if its affine spa…

Hadwiger's theorem concerns common line transversals to families of convex sets in the plane, and it has a generalization to hyperplane transversals in any dimension. A family of c…