I've run into the following problem which I am sure is true but I cannot prove it:
If we color the integers in the set $S = \{1, 2, \ldots, 3n \}$ with $3$ colors such that each color is used $n$ times, there always exist integers $a,b,c \in S$ with distinct colors such that they are in arithmetic progression.
This seems related to Van der Waerden's theorem but we are looking at integers with different colors. Also the fact that we are using $3$ colors $n$ times and are looking at $3n$ integers suggests that this problem may be generalized. Any insights will be appreciated.
In:
Rainbow 3-term Arithmetic Progressions
by Veselin Jungić in Integers, the Electronic Journal of Combinatorial Number Theory, volume 3
we have this abstract:
Consider a coloring of {1, 2,...,n} in 3 colors, where n ≡ 0 (mod 3). If all the color classes have the same cardinality, then there is a 3-term arithmetic progression whose elements are colored in distinct colors. This rainbow variant of van der Waerden’s theorem proves the conjecture of the second author.
Which seems to answer your question. This information is here.