8 problems

For a simple drawing of $K_n$, let $\triangle(n)$ denote the minimum number of empty triangles among all such drawings. An empty triangle is a triangle induced by three vertices su…

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…

A simple drawing of $K_n$ is crossing-maximal if it has $\binom{n}{4}$ crossings; an uncrossed edge is an edge crossed by no other edge. Harborth–Mengersen's conjecture. Every cros…

An empty $k$-cycle in a simple drawing of $K_n$ is a plane cycle through $k$ vertices such that all remaining vertices lie on one common side. Bergold et al.'s conjecture. Every si…

Let $K_n$ be drawn simply in the plane. A plane subdrawing is a subdrawing whose edges do not cross, and a Hamiltonian cycle is a cycle through all $n$ vertices. Bergold et al.'s c…

Let $K_n$ be drawn simply in the plane, and let $a,b$ be two vertices. A path is plane if no two of its edges cross. All-pairs Hamiltonian-path conjecture. For every simple drawing…

Bergold–Fabila-Monroy–Roldán-Pensado–Schaefer conjecture. Every simple drawing of $K_n$ contains an empty $k$-cycle for each $3 leq k leq n$. The supplied paper proves the case…

Unlucky-drawing containment conjecture. Every unlucky drawing contains $\Pi_{8}^{\textnormal{unlucky}}$.