  We consider some coloring issues related to the famous Erd\H {o}s Discrepancy
Problem. A set of the form $A_{s,k}=\{s,2s,\dots,ks\}$, with $s,k\in
\mathbb{N}$, is called a \emph{homogeneous arithmetic progression}. We prove
that for every fixed $k$ there exists a $2$-coloring of $\mathbb N$ such that
every set $A_{s,k}$ is \emph{perfectly balanced} (the numbers of red and blue
elements in the set $A_{s,k}$ differ by at most one). This prompts reflection
on various restricted versions of Erd\H {o}s' problem, obtained by imposing
diverse confinements on parameters $s,k$. In a slightly different direction, we
discuss a \emph{majority} variant of the problem, in which each set $A_{s,k}$
should have an excess of elements colored differently than the first element in
the set. This problem leads, unexpectedly, to some deep questions concerning
completely multiplicative functions with values in $\{+1,-1\}$. In particular,
whether there is such a function with partial sums bounded from above.
