Empty rainbow triangles in k-colored point sets

Monochromatic empty triangles in two-colored point sets

Improving a result of Aichholzer et. al., we show that there exists a constant c > 0 satisfying the following condition. Any two-colored set of n points in general position in the plane has at least cn triples of the same color such that the triangles spanned by them contain no element of the set in their interiors.

On the Number of Empty Pseudo-Triangles in Point Sets

We analyze the minimum and maximum number of empty pseudo-triangles defined by any planar point set. We consider the cases where the three convex vertices are fixed and where they are not fixed. Furthermore, the pseudo-triangles must either be star-shaped or can be arbitrary.

Almost Empty Monochromatic Triangles in Planar Point Sets

Holes or Empty Pseudo-Triangles in Planar Point Sets

Let E(k, l) denote the smallest integer such that any set of at least E(k, l) points in the plane, no three on a line, contains either an empty convex polygon with k vertices or an empty pseudo-triangle with l vertices. The existence of E(k, l) for positive integers k, l ≥ 3, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565–576, 2007]. In this pa...

Rainbow triangles in three-colored graphs

Erdős and Sós proposed a problem of determining the maximum number F (n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F (n) = F (a)+ F (b) +F (c) +F (d) + abc+ abd+ acd+ bcd, where a+ b+ c+ d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4 f...