Rainbow trapezoids with given area

TL;DR

This paper explores monochromatic and rainbow geometric configurations in colored integer lattices, establishing new results on the existence of specific area triangles, rectangles, and trapezoids under various coloring conditions.

Contribution

It introduces novel canonical versions of Euclidean Ramsey results, demonstrating the existence of rainbow and monochromatic polygons with prescribed areas in lattice colorings.

Findings

3-colorings contain a rainbow triangle of area 1/2 or a monochromatic rectangle of any given area

Existence of numbers A and B such that colorings contain either a monochromatic rectangle of area A or a rainbow trapezoid of area B

Results are specific to vertex-colored lattice configurations and geometric shapes

Abstract

A well-known result by Graham in Euclidean Ramsey Theory states that, for every positive real number $A$, every coloring of the plane with finite number of colors contains a monochromatic triangle of area $A$. We consider canonical versions of this result. We show that every $3$-coloring of the plane integer lattice contains either a rainbow triangle of area $1/2$ or a monochromatic rectangle of any given area whose sides are parallell to the axes. We also show that, under natural conditions, there are numbers $A$ and $B$ such that every coloring of the plane integer lattice contains either a monochromatic rectangle of area $A$ or a rainbow trapezoid of area $B$. As usual, only vertex colors are considered: e.g., a monochromatic rectangle is a set of four points in the lattice which a) are the vertices of a rectangle and b) are assigned the same color.