Van der Waerden's theorem

Question

Van der Waerden's theorem states that for any given positive integers $r$ and $k$, there is some number $N$ such that if the integers $\{1, 2, ..., N\}$ are colored, each with one of $r$ different colors, then there are at least k integers in arithmetic progression all of the same color. See wiki example here positive integers from $1$ to $8$ are colored as $BRRBBRRB$, so adding 9th integer as $R$ or $B$ then integer $3, 6, 9$ and $1, 5, 9$ are in AP=3. Here many web/papers refered as monochromatic AP's. What is it meant? also if you color $BRRBBRBB$ then 1, 4, 7 are in AP=3 or if you color $BRRBRRRR$ then $2, 5, 8$ are in AP=3 so you don't need 9th integer as per these arrangement of integer but i guess its not allowed, not sure why. Any enlighten solution will be helpful. Thanks

Answer 1

"monochromatic" simply means "all of the same colour".

The point of the example at https://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem is that you can colour the numbers 1 to 8 with red and blue so that no 3 numbers in arithmetic progression have the same colour - one way of doing this is

BRRBBRRB

But whichever colour we give to 9, we will create a 3 term AP that is either all red or all blue.

Another example is

BBRRBBRR

Now if we colour 9 blue then we have 1-5-9 which are all blue, and if we colour 9 red then we have 7-8-9 which are all red.

This shows that W(2,3) is greater than 8, and suggests (but does not prove) that W(2,3) is actually 9.

There are, of course, many colourings of 1 to 8 which have a 3 term AP that is all the same colour - for example, any colouring starting BBB... has 1-2-3 which are all blue. The point of the example is that there are some colourings of 1 to 8 which avoid this.