Coloring Subgraphs of the Rado Graph

Given a universal binary countable homogeneous structure U and n ∈ ω, there is a partition of the induced n-element substructures of U into finitely many classes so that for any partition C0, C1, . . . , Cm−1 of such a class Q into finitely many parts there is a number k ∈ m and a copy U∗ of U in U so that all of the induced n-element substructures of U∗ which are in Q are also in Ck. The partition of the induced n-element substructures of U is explicitly given and a somewhat sharper result as the one stated above is proven.