Empty Monochromatic Triangles

Almost Empty Monochromatic Triangles in Planar Point Sets

Contents Combinatorial Convexity by Víctor Álvarez and Jeong

Erdős-Szekeres theorem is one of classic results in combinatorial geometry. In this project we consider the colored version of the problem. Especially, we are interested in the number of empty monochromatic triangles and the existence of an empty monochromatic convex quadrilateral. We give some minor results and plausible ideas to solve the problems.

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.

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...

The Maximum Number of Empty Congruent Triangles Determined by a Point Set

Let be a set of points in the plane and consider a family of (nondegenerate) pairwise congruent triangles whose vertices belong to . While the number of such triangles can grow superlinearly in — as it happens in lattice sections of the integer grid — it has been conjectured by Brass that the number of pairwise congruent empty triangles is only at most linear. We disprove this conjecture by con...