Let .
Then there is some such that whenever is -coloured,
there exists a monochromatic arithmetic progression of length ,
whose difference is of the same colour.
Proof
Let be suitable for .
We show that the Waerden’s Number is suitable for .
By definition, take monochromatic.
If any of are the same colour as this sequence, we are done.
Otherwise, this sequence is -coloured,
so we are done by induction hypothesis.