Monochromatic configurations for finite colourings of the plane


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.

DOI Code: 10.1285/i15900932v22n1p59

Keywords: Gurevich's conjecture; Monochromatic configurations

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