9.5. Ramsey theorem in Number Theory

The Problem: Let us colour the numbers between 1 and 9 with blue and red. Are there always 3 blue or red numbers among them which form a numerical sequence?

Case 1. Let us suppose that the numbers 4 and 6 has the same colour (blue).

We need to avolid the numerical sequence 4, 5, 6 so we must colour 5 with red.

Now, we need to avoid the 2, 4, 6 blue numerical sequence and the 4, 6, 8 numerical sequence. Therefore we need to colour 2 and 8 with red.

But now the numbers 2, 5, 8 form a red numerical sequence.

Case 2. Let 4 and 6 be coloured with different colours. (4 red and 6 blue.)

Then 5 may have any colour we do not get a numerical sequence. Let 5 be coloured with red.

Now, we need to continue the colouring in the following way:

3 by blue to avoid the 3, 4, 5 red sequence.

9 by red to avoid the 3, 6, 9 blue sequence.

7 by blue to avoid the 5, 7, 9 red sequence.

8 by red to avoid the 6, 7, 8 blue sequence.

2 by blue to avoid the 2, 5, 8 red sequence.

1 by red to avoid the 1, 2, 3 blue sequence.

But now we have a red numerical sequence 1, 5, 9.

Theorem 9.11. (Van der Waerden, 1926): There exits such integer for wich if then any k-colouring of the set contains a monochromatic numerical subsequent with h elements.