9 problems

A $k$-crossing family is a set of $k$ independent edges that cross pairwise. A simple drawing of $K_n$ is crossing-maximal when it has the maximum possible number $\binom{n}{4}$ of…

Strengthening-set partition conjecture. If

Let $S$ be a set of $n$ points in the plane in general position. An elbow is an orthogonal geometric-graph edge consisting of one horizontal and one vertical line segment, and a cr…

Let $S$ be a set of $n$ points in the plane in general position. A crossing matching is a matching whose every pair of edges crosses. Aronov et al.'s conjecture. Every such point s…

Let $S$ be a set of $n$ points in the Euclidean plane in general position, meaning that no three points of $S$ are collinear. A crossing family is a set of segments spanned by poin…

Let $S$ be a set of $n$ points in the Euclidean plane in general position, meaning that no three points of $S$ are collinear. A segment of $S$ is a line segment whose endpoints are…

Linear pairwise-crossing segments conjecture. The function $T(n)$ should satisfy

A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. For integers $k,l\geq 1$, a $(k,l)$-crossing family is a pa…

Let $t(n)$ be the minimum number of colours needed to colour the edges of some geometric drawing of $K_n$ so that every pair of edges with the same colour crosses; each colour clas…