Let .
Then there is some such that whenever is -coloured,
there exists a monochromatic Combinatorial Line.
Write for the smallest such .
Proof
By induction on .
Given , assume exists for all .
Suppose there is no monochromatic line in for any
We prove for all there is some
such that contains Colour-Focused lines.
For set .
Given suitable for , let .
Identify with .
There are ways to colour a copy of .
By choice of , there is a monochromatic line in ,
so extend it to a line in .
Now any has a colour given by this line
(i.e. for any , the colour of is the same as of ).
Thus we colour in colours induced by the line .
By the choice of , there are Colour-Focused lines in ,
under this new colouring, say focused at .
This gives Colour-Focused lines in .
Moreover, the line is also focused at .
But is of a different colour from ,
otherwise we would’ve had a monochromatic line in .
This completes the -induction, and the claim follows.