Let .
Then there is some
such that whenever is -coloured,
there is a monochromatic set .
Proof
Suppose not.
For each , there is a colouring
with no monochromatic -set.
Now there are only finitely many colourings of ,
so for each , infinitely many of must agree on .
Thus construct a sequence of colourings
such that and agree on ,
and any doesn’t have a monochromatic -set.
Finally, take the limiting colouring
and apply Ramsey’s Theorem.