Let .
Then there is some such that whenever is -coloured,
there exists a monochromatic arithmetic progression of length .
In particular, the Waerden’s Number exists.
Proof
We proceed by induction on .
The case is trivial.
Suppose we have proved the claim for ,
and assume there is no monochromatic progression of length .
We show the following claim:
For all , there exists some such that whenever is -coloured,
there exist Colour-Focused arithmetic progressions of length .
We do an induction on .
For , we can set .
Now assume that is suitable for .
We claim that is suitable for .
Split into blocks of size .
There is ways to colour a block,
so we can find identically coloured.
Now the first half of has Colour-Focused progressions
of length
As the length of is , the focus of these progressions is also in .
Now consider the progressions .
They are Colour-Focused at .
But so is , which is also monochromatic,
but is a different colour from all the other ones.
Thus we found the Colour-Focused progressions.
This completes the proof, as applying the claim for gives a contradiction.