Prove that there exists a four-coloring of the set
![M = \{1, 2, \cdots, 1987\}](/media/m/5/7/a/57aa7942d3a5d9112114ac2b65a6ee37.png)
such that any arithmetic progression with
![10](/media/m/5/b/e/5beb46430dbe2d22c0f8289c36a92c84.png)
terms in the set
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
is not monochromatic.
Alternative formulation
Let
![M = \{1, 2, \cdots, 1987\}](/media/m/5/7/a/57aa7942d3a5d9112114ac2b65a6ee37.png)
. Prove that there is a function
![f : M \to \{1, 2, 3, 4\}](/media/m/f/7/8/f78c8791dc5ac6ecf1296133b6a1fba3.png)
that is not constant on every set of
![10](/media/m/5/b/e/5beb46430dbe2d22c0f8289c36a92c84.png)
terms from
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
that form an arithmetic progression.
Proposed by Romania
%V0
Prove that there exists a four-coloring of the set $M = \{1, 2, \cdots, 1987\}$ such that any arithmetic progression with $10$ terms in the set $M$ is not monochromatic.
Alternative formulation
Let $M = \{1, 2, \cdots, 1987\}$. Prove that there is a function $f : M \to \{1, 2, 3, 4\}$ that is not constant on every set of $10$ terms from $M$ that form an arithmetic progression.
Proposed by Romania