Patterns and complexity of multiplicative functions

I. Schur and G. Schur proved that, for all completely multiplicative functions f : N -> {-1, 1}, with the exception of two character-like functions, there is always a solution of f(n) = f(n + 1) = f(n + 2) = 1. Hildebrand proved that for the Liouville lambda-function each of the eight possible sign combinations (lambda(n),lambda(n + 1),lambda(n + 2)) occurs infinitely often. We prove for completely multiplicative functions f : N -> {-1, 1}, satisfying certain necessary conditions, that any sign pattern (epsilon(1), epsilon(2), epsilon(3), epsilon(4)), epsilon(i) is an element of {-1, 1}, occurs for infinitely many 4-term arithmetic progressions (f(n), f(n + d), f(n + 2d), f(n + 3d)). The proof introduces graph theory and new combinatorial methods to the subject.