Question

on a 2d complex plane, all the integer-component points are coloured either white or black. is possible to find a rectangle parallel to axis which has all corners of same color?

Answer 1

I have a 3-coloring of Z×Z, i.e. a function f:Z×Z→{red,green,blue}.

I have to prove that there is a monochromatic rectangle with its sides being parallel to the axis, i.e. to prove that for some choice of a,b,c,d∈Z with a≠b and c≠d, all the points (a,c),(a,d),(b,c),(b,d) have the same color.

I tried to work by contradiction, without achieving much.

Additionally, can we prove some upper bound on |a−b| and |c−d|?