Empty monochromatic triangles

Empty Monochromatic Triangles

We consider a variation of a problem stated by Erdös and Guy in 1973 about the number of convex k-gons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As a main result we show that any bi-colored set of n points in R in general position determines a ...

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.

Almost Empty Monochromatic Triangles in Planar Point Sets

Empty Monochromatic Simplices

Let S be a k-colored (finite) set of n points in R, d ≥ 3, in general position, that is, no (d+1) points of S lie in a common (d−1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3 ≤ k ≤ d we provide a lower bound of Ω(nd−k+1+...

Monochromatic triangles in three-coloured graphs

In 1959, Goodman [9] determined the minimum number of monochromatic triangles in a complete graph whose edge set is 2-coloured. Goodman [10] also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, for n sufficiently large, we determine the minimum number of monochromatic triangles in a 3-coloured copy of K...