Monochromatic configurations for finite colourings of the plane

Abstract

A strengthened form of Gurevich's conjecture was proved by R.L.Graham ([2],[3]),Which says that for any $\alpha >0$ and any pair of non-parallel lines $L_1$ and $L_2$, in any partition of the plane into finitely many classes, some class contains the vertices of a triangle which has area $\alpha$ and two sides parallel to the lines $L_i$.  Later, a shorter proof, using the main idea of Graham, was presented in [1]. Following some questions raised by Graham [2] and by suitable modifications of methods therein, here we establish a similiar in the case of vertices of a trapezium.