MathDB

Let

E

denote the set of

25

points

(m,n)

in the

\text{xy}

-plane, where

m,n

are natural numbers,

1\leq m\leq5,1\leq n\leq5

. 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 the axes and having at least three points of the same colour.)

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