2. Sign patterns

Problem 2.1.
Let $f$ be a multiplicative function with countable range. Can we find a coloring of that range with (say) two colors, such that the number of "sign patterns" of $f$ (in terms of the colors) grows superlinearly? 
Problem 2.2.
Can we show arbitrarily large patterns $++\ldots+$ or $\ldots $ in $(\lambda(n))_{n \in \N}$? An approach through the entropy decrement argument (roughly) reduces this to showing the existence of arbitrarily large progressions $n, n+p, \dots, n+kp$ with $\lambda(n) = \lambda(n+p) = \dots = \lambda(n+kp)$.
Cite this as: AimPL: Sarnak's conjecture, available at http://aimpl.org/sarnakconjecture.