Let E denote the set of 25 points (m,n) in the xy-plane, where m,n are natural numbers, 1 ≤ m ≤ 5, 1 ≤ n ≤ 5. Suppose the points of E are arbitrarily coloured using two colours, red and blue. Show that there always exist four points in the set E of the form (a,b), (a + k,b), (a + k, b + k), (a,b + k) for some positive integer k such that at least three of these four points have the same colour. (That is, there always exist four points in the set E which form the vertices of a square with sides parallel to axes and having at least three points of the same colour.)
