Simple proof that W(3,3)=27?

Question

I was wondering that if there exists a simple proof that the van der Warden's number W(3,3) (the smallest positive integer $n$ such that any 3-coloring of the set $\{1, 2, ..., n\}$ has a monochromatic 3-AP) equals 27. Also, it would be great if someone can present me a counterexample for $n=26$.

Answer 1

The complete list of length-26 examples is:

    AABBAABCBCCACAABABBCACCBCB
    AABBCAACACBBCCBBCACAACBBAA
    AABCBAACACBBCCBBCACAABCBAA
    AACBCAABABCCBBCCBABAACBCAA
    AACCAACBCBBABAACACCBABBCBC
    AACCBAABABCCBBCCBABAABCCAA
    ABAABABCCBBCCBABAABCCAACAC
    ABABBAACBBCBCAACCAACBCBBCA

not counting relabeling of the colors. (So there are 48 examples in all.)

I don't know of any proof that 27 is exact, other than a suitably-pruned exhaustive search.

I have source code that I wrote about 25 years ago that does the search, plus some discussion, on my blog. The code took a couple of days to run when I first wrote it, but to run it now is very quick.