Davies’s theorem and the coloring number of graphs

We generalize Davies’s famous example to a property of graphs: a graph ( V , X ) (V,X) is Davies if for every F : X → R F:X→ R , there is a collection { g v : v ∈ V } \{g_v:v∈ V\} of ω → R ω → R functions such that F ( v , w ) = ∑ n > ω g v ( n ) g w ( n ) F(v,w)=∑ n>ω g_v(n)g_w(n) for { v , w } ∈ X \{v,w\}∈ X . We show that if C o l ( X ) ≤ ω 1 Col(X)≤ ω _1 and | X |