Let $m \ge 2 ,n \ge 2$ be positive integers which are coprime (that means that the greatest common divisor of $m,n$ is $1$). Is it possible to paint the set $\mathbf{N}$ of all natural numbers using $4$ colours such that there are no positive integers $a,b,c,d$ of the same colour satisfying $ma+mb = nc+nd?$

The answer is YES for the specific case when $m=3, n=2.$ For instance if $h \in \mathbf{N}$ is of the form $h = 5^m (5k+i)$ for $i \in \{1,2,3,4 \}$ then if we paint $h$ using the $i$ th color then by chasing the coefficient of $i$ it is easy to see that we do not have a monochromatic solution to the equation $3a+3b=2c+2d.$

In fact if one goes through the proof for this particular case then it seems to imply a much weaker relation where $4$ gets replaced by a larger number of colors.

5more comments