Rainbow trapezoids with given area

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.