 Solution

Let $\rho$ be the cyclic logarithmic coloring from part (iii). Define a finite coloring of the positive integers byThe inner logarithm is at least $1$ whenever $n\ge 2$, so this is defined everywhere. Suppose an increasing infinite sequence $(x_i)$ made every product with distinct indices monochromatic. Fix an index $i$ with $x_i\ge 2$ and write $a={\log }_2{x}_i>0$. The sequence is unbounded, so choose $j>i$ with $b={\log }_2{x}_j\ge 28a$. Both ordered products are among the supposedly monochromatic values, but their logarithms satisfyIndeed, the first inequality is equivalent to $b\ge 28a$, and the second uses $a>0$. Part (iii) therefore gives $\chi (x_i{x}_j^2)\ne \chi (x_j{x}_i^2)$, a contradiction. Thus the assertion with all distinct ordered indices is false. The obstruction needs an infinite unbounded sequence; it does not assert that every pair of different positive integers gives different colors.