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 λ‐function each of the eight possible sign combinations (λ( n ), λ( n +1), λ( n +2)) occurs infinitely often. We prove for completely multiplicative functions f :ℕ→{−1, 1}, satisfying certain necessary conditions, that any sign pattern (ε 1 , ε 2 , ε 3 , ε 4 ), ε i ∈{−1, 1}, occurs for infinitely many 4‐term arithmetic progressions( f ( n ) ) , f ( n + d ) , f ( n + 2 d ) , f ( n + 3 d ) .The proof introduces graph theory and new combinatorial methods to the subject.