5 problems

Let $R$ and $B$ be disjoint sets of $n$ red and $n$ blue points, respectively, in the plane. An empty red-red-blue triangle is a triangle whose vertices consist of two points of…

Let $K_n$ be the complete graph on $n$ vertices, and let a good drawing of $K_n$ be a drawing in which edges meet only at common endpoints or proper crossings, with no three edges…

Let $D(K_n)$ be a good drawing of the complete graph $K_n$ with vertex set of size $n$. An empty triangle is a triangular face of the drawing whose boundary consists of three edges…

Empty-triangle counting conjecture. For large enough $n$, $f(n)>1.01n^2$. The conjecture concerns the minimum possible number of empty triangles in a planar point set. The source r…

Bárány's conjecture. As $n\to\infty$, $\deg X$ goes to infinity. This conjecture asks whether every sufficiently large finite planar point set contains a pair belonging to arbitrar…